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## why math is indispensable short essay

Mathematics is a subject that is often seen as difficult, but it’s actually one of the most essential subjects for students to learn. In this article, we’re going to explore why mathematics is so important and how it can be incredibly useful in your everyday life.

## Mathematics is indispensable in fields such as engineering, medical science and the law.

Math is a logical and rational process that can be easily applied to solve problems. It allows for predictions and understanding of patterns. Mathematics is essential in the field of engineering because it allows for the design of efficient and effective machines and structures. It also allows for the prediction of how something will behave under various environmental conditions. In medicine, mathematics allows for the understanding of how organs and systems work. It is also used to determine the effectiveness of treatments. Finally, mathematics plays an important role in the law. Law relies heavily on mathematical models in order to make predictions about how people will behave and what actions will be taken.

## Mathematics has applications in virtually all aspects of life.

Some of the many reasons why mathematics is so important include:

-Mathematics is essential for solving problems.

-Mathematics can help us understand the natural world and how it works.

-Mathematics can help us design things better.

-Mathematics is essential for understanding economics and financial systems.

-Mathematics is essential for planning and predicting the future.

-Mathematics can help us understand the human brain and how it works.

## Mathematics allows us to understand patterns and relationships that we could never see otherwise.

Mathematics is the foundation of all scientific inquiry, and its concepts are essential for understanding how the world works. Mathematics has helped us develop technologies that we use every day, from GPS to electronic commerce. Mathematics has even played a role in helping us understand the universe itself.

There are countless reasons why mathematics is so important, and it is impossible to list them all here. But hopefully this essay has given you a flavor for what mathematics can do for us and showed just how indispensable it is in our everyday lives.

## Mathematics has a deep impact on our daily lives, ranging from making financial decisions to solving problems related to physics and chemistry.

Mathematics is indispensable in a world where we rely on computers and other technology to do our work. In fact, without mathematics, it would be difficult or impossible to develop modern technology. For example, when engineers design computer circuits, they need to know about the mathematical properties of numbers such as addition, subtraction, multiplication, and division. Without knowing these mathematical principles, it would be very difficult to create even the simplest circuit.

Math also has a profound impact on physics and chemistry. Scientists use mathematics to study the behavior of matter and energy at the atomic and subatomic level. This knowledge is essential for understanding how the world works and for developing new technologies. In fact, without mathematics, scientists would not be able to build electron microscopes or understand radioactive decay.

So what does all this have to do with you? Plenty! The skills you learn in mathematics—including solving problems, thinking critically, and working out solutions—are essential for success in any field. If you want to be a doctor, scientist, mathematician, or engineer, you need to start learning basic mathematics right away.

## Without mathematics, many technological advances would not be possible.

Mathematics is essential for technologies such as airplanes, computers, and cellphones. Without mathematics, these devices would not be able to function or transmit information. Mathematics is also essential for medical treatments and scientific research.

Mathematicians are constantly working on new ways to use mathematics in technology and research. They are also working on creating new mathematical concepts that can be used in technology and research. Mathematics is a very versatile field that can be used in many different areas of life.

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## Essay on Importance of Mathematics in our Daily Life in 100, 200, and 350 words.

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- Dec 22, 2023

Mathematics is one of the core aspects of education. Without mathematics, several subjects would cease to exist. It’s applied in the science fields of physics, chemistry, and even biology as well. In commerce accountancy, business statistics and analytics all revolve around mathematics. But what we fail to see is that not only in the field of education but our lives also revolve around it. There is a major role that mathematics plays in our lives. Regardless of where we are, or what we are doing, mathematics is forever persistent. Let’s see how maths is there in our lives via our blog essay on importance of mathematics in our daily life.

## This Blog Includes:

Essay on importance of mathematics in our daily life in 100 words , essay on importance of mathematics in our daily life in 200 words, essay on importance of mathematics in our daily life in 350 words.

Mathematics is a powerful aspect even in our day-to-day life. If you are a cook, the measurements of spices have mathematics in them. If you are a doctor, the composition of medicines that make you provide prescription is made by mathematics. Even if you are going out for just some groceries, the scale that is used for weighing them has maths, and the quantity like ‘dozen apples’ has maths in it. No matter the task, one way or another it revolves around mathematics. Everywhere we go, whatever we do, has maths in it. We just don’t realize that. Maybe from now on, we will, as mathematics is an important aspect of our daily life.

Also Read:- Importance of Internet

Mathematics, as a subject, is one of the most important subjects in our lives. Irrespective of the field, mathematics is essential in it. Be it physics, chemistry, accounts, etc. mathematics is there. The use of mathematics proceeds in our daily life to a major extent. It will be correct to say that it has become a vital part of us. Imagining our lives without it would be like a boat without a sail. It will be a shock to know that we constantly use mathematics even without realising the same.

From making instalments to dialling basic phone numbers it all revolves around mathematics.

Let’s take an example from our daily life. In the scenario of going out shopping, we take an estimate of hours. Even while buying just simple groceries, we take into account the weight of vegetables for scaling, weighing them on the scale and then counting the cash to give to the cashier. We don’t even realise it and we are already counting numbers and doing calculations.

Without mathematics and numbers, none of this would be possible.

Hence we can say that mathematics helps us make better choices, more calculated ones throughout our day and hence make our lives simpler.

Also Read:- My Aim in Life

Mathematics is what we call a backbone, a backbone of science. Without it, human life would be extremely difficult to imagine. We cannot live even a single day without making use of mathematics in our daily lives. Without mathematics, human progress would come to a halt.

Maths helps us with our finances. It helps us calculate our daily, monthly as well as yearly expenses. It teaches us how to divide and prioritise our expenses. Its knowledge is essential for investing money too. We can only invest money in property, bank schemes, the stock market, mutual funds, etc. only when we calculate the figures. Let’s take an example from the basic routine of a day. Let’s assume we have to make tea for ourselves. Without mathematics, we wouldn’t be able to calculate how many teaspoons of sugar we need, how many cups of milk and water we have to put in, etc. and if these mentioned calculations aren’t made, how would one be able to prepare tea?

In such a way, mathematics is used to decide the portions of food, ingredients, etc. Mathematics teaches us logical reasoning and helps us develop problem-solving skills. It also improves our analytical thinking and reasoning ability. To stay in shape, mathematics helps by calculating the number of calories and keeping the account of the same. It helps us in deciding the portion of our meals. It will be impossible to think of sports without mathematics. For instance, in cricket, run economy, run rate, strike rate, overs bowled, overs left, number of wickets, bowling average, etc. are calculated. It also helps in predicting the result of the match. When we are on the road and driving, mathetics help us keep account of our speeds, the distance we have travelled, the amount of fuel left, when should we refuel our vehicles, etc.

We can go on and on about how mathematics is involved in our daily lives. In conclusion, we can say that the universe revolves around mathematics. It encompasses everything and without it, we cannot imagine our lives.

Also Read:- Essay on Pollution

Ans: Mathematics is a powerful aspect even in our day-to-day life. If you are a cook, the measurements of spices have mathematics in them. If you are a doctor, the composition of medicines that make you provide prescription is made by mathematics. Even if you are going out for just some groceries, the scale that is used for weighing them has maths, and the quantity like ‘dozen apples’ has maths in it. No matter the task, one way or another it revolves around mathematics. Everywhere we go, whatever we do, has maths in it. We just don’t realize that. Maybe from now on, we will, as mathematics is an important aspect of our daily life.

Ans: Mathematics, as a subject, is one of the most important subjects in our lives. Irrespective of the field, mathematics is essential in it. Be it physics, chemistry, accounts, etc. mathematics is there. The use of mathematics proceeds in our daily life to a major extent. It will be correct to say that it has become a vital part of us. Imagining our lives without it would be like a boat without a sail. It will be a shock to know that we constantly use mathematics even without realising the same.

From making instalments to dialling basic phone numbers it all revolves around mathematics. Let’s take an example from our daily life. In the scenario of going out shopping, we take an estimate of hours. Even while buying just simple groceries, we take into account the weight of vegetables for scaling, weighing them on the scale and then counting the cash to give to the cashier. We don’t even realise it and we are already counting numbers and doing calculations.

Without mathematics and numbers, none of this would be possible. Hence we can say that mathematics helps us make better choices, more calculated ones throughout our day and hence make our lives simpler.

Ans: Archimedes is considered the father of mathematics.

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Home — Essay Samples — Science — Mathematics in Everyday Life — Mathematics In Everyday Life: Most Vital Discipline

## Mathematics in Everyday Life: Most Vital Discipline

- Categories: Mathematics in Everyday Life

About this sample

Words: 795 |

Published: Mar 14, 2019

Words: 795 | Pages: 2 | 4 min read

## Works Cited

- Benacerraf, P. (1991). Mathematics as an object of knowledge. In P. Benacerraf & H. Putnam (Eds.), Philosophy of mathematics: Selected readings (pp. 1-13). Cambridge University Press.
- EdReady. (n.d.). Home. Retrieved from https://www.edready.org/
- Puttaswamy, T. K. (2012). Engineering mathematics. Dorling Kindersley (India) Pvt. Ltd.
- Steen, L. A. (Ed.). (2001). Mathematics today: Twelve informal essays. Springer Science & Business Media.
- Suter, B. W. (2012). Mathematics education: A critical introduction. Bloomsbury Academic.
- Tucker, A. W. (2006). Applied combinatorics. John Wiley & Sons.
- Vakil, R. (2017). A mathematical mosaic: Patterns & problem solving. Princeton University Press.
- Wolfram MathWorld. (n.d.). MathWorld--The web's most extensive mathematics resource. Retrieved from http://mathworld.wolfram.com/
- Wu, H. H. (2011). The mis-education of mathematics teachers. Educational Studies in Mathematics, 77(1), 1-20.
- Ziegler, G. M., & Aigner, M. (2012). Proofs from THE BOOK. Springer Science & Business Media.

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## Essay on Importance of Mathematics in Our Daily Life

Students are often asked to write an essay on Importance of Mathematics in Our Daily Life in their schools and colleges. And if you’re also looking for the same, we have created 100-word, 250-word, and 500-word essays on the topic.

Let’s take a look…

## 100 Words Essay on Importance of Mathematics in Our Daily Life

Introduction.

Mathematics is a crucial part of everyday life. It helps us make sense of the world around us and solve practical problems.

## Mathematics in Daily Tasks

From shopping to cooking, we use math. It helps us calculate costs, quantities, and time.

## Mathematics in Professions

In professions like engineering, computer science, and finance, math is indispensable.

## Mathematics in Decision Making

Math helps us make informed decisions by analyzing data and predicting outcomes.

Thus, math plays a vital role in our daily lives, making it an essential subject to learn.

## 250 Words Essay on Importance of Mathematics in Our Daily Life

The pervasive presence of mathematics.

Mathematics, often perceived as a complex and abstract discipline, is in fact an integral part of our everyday lives. It forms the foundation for many of the decisions we make and the actions we perform daily, from managing finances to navigating directions.

## A Tool for Logical Reasoning

Mathematics fosters logical reasoning and problem-solving skills. It cultivates an analytical mindset, enabling us to break down complex problems into simpler, manageable parts. This approach is not just confined to mathematical problems but extends to various real-life situations, thereby honing our decision-making abilities.

## Mathematics in Technological Advancements

The rapid progress in technology, which has become an inseparable part of our lives, is deeply rooted in mathematical principles. Algorithms, which form the basis of computing, are mathematical models. The internet, smartphones, GPS, and even AI owe their existence to mathematical concepts.

## Financial Management and Mathematics

Managing personal finances, a critical life skill, is essentially a mathematical exercise. Budgeting, calculating interest, understanding the implications of loans and mortgages, or even evaluating investment options, all require a good grasp of mathematics.

## Mathematics and Scientific Understanding

Mathematics is the language of science. It helps us quantitatively understand and describe the physical world around us, from the trajectory of planets to the behavior of subatomic particles.

In conclusion, mathematics is a vital part of our daily lives. It is not just a subject to be studied in school, but a tool for understanding, navigating, and shaping the world around us. Its importance cannot be overstated, as it is the foundation of critical thinking, technological progress, financial management, and scientific understanding.

## 500 Words Essay on Importance of Mathematics in Our Daily Life

Mathematics, often perceived as a complex and abstract subject, is in fact deeply intertwined with our daily lives. It is the foundation of numerous activities we engage in, from basic tasks such as shopping and cooking to more complex ones like planning finances or solving problems.

## The Ubiquity of Mathematics

Mathematics is everywhere. It is used in our everyday activities, often without our conscious realization. When we shop, we use mathematics to calculate prices, discounts, and taxes. When we cook, we use it to measure ingredients. When we travel, we use it to calculate distances, time, and fuel consumption. Even in our leisure activities such as playing games or music, mathematics plays a crucial role in understanding patterns, probabilities, and rhythms.

## Mathematics in the Professional Sphere

In the professional world, the significance of mathematics is even more pronounced. Engineers use mathematical principles to design and build infrastructure. Economists use it to predict market trends. Computer scientists use algorithms and data structures, which are fundamentally mathematical in nature, to design efficient software. Even in fields that are traditionally considered non-mathematical, such as literature and arts, mathematical concepts such as symmetry, geometry, and proportion play a key role in creating aesthetic appeal.

## Mathematics and Problem-Solving

Mathematics also enhances our problem-solving skills. It teaches us to approach problems logically and systematically. It encourages us to break down complex problems into simpler parts, solve them individually, and combine the solutions to solve the original problem. This skill is not just applicable to mathematical problems but to any problem we encounter in life.

## Mathematics and Critical Thinking

Furthermore, mathematics fosters critical thinking. It trains us to question assumptions, identify patterns, and draw conclusions based on evidence. It also teaches us to understand the limitations of our solutions and consider alternative approaches. These are valuable skills that can be applied in various aspects of life, from making informed decisions to evaluating the credibility of information.

## Mathematics and the Digital Age

In the digital age, the importance of mathematics has grown exponentially. It is the backbone of modern technologies such as artificial intelligence, machine learning, data analysis, cryptography, and quantum computing. Understanding mathematics is essential to navigate and thrive in this digital world.

In conclusion, mathematics is not just a subject we learn in school. It is a powerful tool that helps us understand and navigate the world around us. It enhances our problem-solving and critical thinking skills, and it opens up a world of opportunities in the professional sphere. Therefore, it is essential that we appreciate the importance of mathematics in our daily lives, and strive to improve our mathematical literacy.

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## Why Math Is Important: A Student’s View (Updated for 2024)

A couple of weeks ago, I asked my son to write an extra essay for a project we were working on for the Classical Conversations Practicum . I allowed him to work on that essay instead of his math lesson for the day.

Suddenly, my daughter, Ada, did not want to do her math lesson for the day. I explained that my son was writing an essay instead and she asked to be allowed to do the same.

I thought, “Hmm… This is a good time for my daughter to think about why math is important and come up with her own reasons for studying this subject.” She wrote the following essay and we thought it might help parents and students to read Ada’s thoughts.

Pure mathematics is, in its way, the poetry of logical ideas.” – Albert Einstein

## “Why Is Math Important?” by Ada Bianco

“Everyone agrees that learning math can be difficult, but some people believe math is important and some people believe math is not important. Math is important for three reasons:

- Math is everywhere.
- Children need math.
- God created math.

## Math Is Everywhere

The first reason math is important is because it is everywhere. It is used in everyday life. It is useful, but it is more than just useful. Math is there to help us, to keep us well ordered, to help us learn new things and to help teach us new things. Students will become adults who will use math in their jobs. All kinds of careers use math. For example, musicians, accountants, fashion designers, and mothers use math. However, math is not only used for things you do. It also brings order to everything around you. The world is organized essentially because it was made with math.

## Children Need Math

The second reason math is important is children need math. Now, as we all know, children are as chaotic as a volcanic eruption. But as they grow, children need to learn patience. Patience is precisely what math teaches us. It also teaches us curiosity. For example, why is this rule used here? Why would that number be negative? Why is that equation set up like that? These are questions they will learn to ask if they are taught math. The parents’ job is to help their children grow up to become good people who are patient and wise, who want to learn even more about anything and everything. Their future depends on what they have learned and if they have learned mathematics, then they will be able to do many different things—maybe even anything—when they are adults

## God Created Math

The third reason math is important is God created it. This is a reason most adults use to convince their children that math is not boring and unimportant, so it may seem unoriginal. I believe, however, it is something that needs to be stated. God created the universe as well as math. The universe is full of math and it is orderly because of math. The sun is a certain distance from the earth; everything is organized in such a way that no matter what has happened we have always been safe. We need math. From this, you should be able to see how much we really do use and need math. We would not be able to process or even do everyday things without it. Math, in addition to these things, helps us to know God. God gave us math to live well and to serve Him. With everything we learn using math in science, we learn more about the world, which can help draw us closer to God.

Some people say math is unimportant because you don’t need math other than basic math principles—you can live without more complicated math. They say, if you need it, then simply use a calculator and leave the more complicated math to people who like math, the mathematicians. This, however, is not correct. You need math and could not live well without math, even including more complicated math concepts. God made us with a sense of curiosity so we can learn, do, and think about all sorts of things. Math is that thing that connects everything together, everything people love to do: music, cooking, painting, and everything else. Math is important.

Math is important because math is everywhere, children need math, and God created math. This matters to me and other children because math determines our future and how we choose to live.”

## Understanding Why We Learn Subjects

Often, it’s not just our students who struggle to understand why math is important. Some of us can use a reminder as well. Hopefully Ada’s essay inspired you with reasons why we and our students should learn math. What students learn from math, just as with studying any other subject, can be applied to all areas of life. Having your student write a persuasive essay like the one above can be a great way to help them understand how a subject they are studying is useful in everyday life.

Of course, your student’s topic doesn’t have to be math. It can be history, geography, English, literature, Latin, science, or any other subject. Whatever subject they write about, this exercise is undeniably useful for helping them to understand the point of studying that subject. Perhaps they’ll even surprise you with reasons you haven’t thought of.

-Ada Bianco

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## Indispensability Arguments in the Philosophy of Mathematics

One of the most intriguing features of mathematics is its applicability to empirical science. Every branch of science draws upon large and often diverse portions of mathematics, from the use of Hilbert spaces in quantum mechanics to the use of differential geometry in general relativity. It’s not just the physical sciences that avail themselves of the services of mathematics either. Biology, for instance, makes extensive use of difference equations and statistics. The roles mathematics plays in these theories is also varied. Not only does mathematics help with empirical predictions, it allows elegant and economical statement of many theories. Indeed, so important is the language of mathematics to science, that it is hard to imagine how theories such as quantum mechanics and general relativity could even be stated without employing a substantial amount of mathematics.

From the rather remarkable but seemingly uncontroversial fact that mathematics is indispensable to science, some philosophers have drawn serious metaphysical conclusions. In particular, Quine (1976; 1980a; 1980b; 1981a; 1981c) and Putnam (1979a; 1979b) have argued that the indispensability of mathematics to empirical science gives us good reason to believe in the existence of mathematical entities. According to this line of argument, reference to (or quantification over) mathematical entities such as sets, numbers, functions and such is indispensable to our best scientific theories, and so we ought to be committed to the existence of these mathematical entities. To do otherwise is to be guilty of what Putnam has called “intellectual dishonesty” (Putnam 1979b, p. 347). Moreover, mathematical entities are seen to be on an epistemic par with the other theoretical entities of science, since belief in the existence of the former is justified by the same evidence that confirms the theory as a whole (and hence belief in the latter). This argument is known as the Quine-Putnam indispensability argument for mathematical realism. There are other indispensability arguments, but this one is by far the most influential, and so in what follows, we’ll mostly focus on it.

In general, an indispensability argument is an argument that purports to establish the truth of some claim based on the indispensability of the claim in question for certain purposes (to be specified by the particular argument). For example, if explanation is specified as the purpose, then we have an explanatory indispensability argument. Thus we see that inference to the best explanation is a special case of an indispensability argument. See the introduction of Field (1989, pp. 14–20) for a nice discussion of indispensability arguments and inference to the best explanation. See also Maddy (1992) and Resnik (1995a) for variations on the Quine-Putnam version of the argument. We should add that although the version of the argument presented here is generally attributed to Quine and Putnam, it differs in a number of ways from the arguments advanced by either Quine or Putnam. [1]

## 1. Spelling Out the Quine-Putnam Indispensability Argument

2. what is it to be indispensable, 3. naturalism and holism, 4. objections, 5. explanatory versions of the argument, 6. conclusion, other internet resources, related entries.

The Quine-Putnam indispensability argument has attracted a great deal of attention, in part because many see it as the best argument for mathematical realism (or platonism). Thus anti-realists about mathematical entities (or nominalists) need to identify where the Quine-Putnam argument goes wrong. Many platonists, on the other hand, rely very heavily on this argument to justify their belief in mathematical entities. The argument places nominalists who wish to be realist about other theoretical entities of science (quarks, electrons, black holes and such) in a particularly difficult position. For typically they accept something quite like the Quine-Putnam argument [2] ) as justification for realism about quarks and black holes. (This is what Quine (1980b, p. 45) calls holding a “double standard” with regard to ontology.)

For future reference, we’ll state the Quine-Putnam indispensability argument in the following explicit form:

(P1) We ought to have ontological commitment to all and only the entities that are indispensable to our best scientific theories. (P2) Mathematical entities are indispensable to our best scientific theories. (C) We ought to have ontological commitment to mathematical entities.

Thus formulated, the argument is valid. This forces the focus onto the two premises. In particular, a couple of important questions naturally arise. The first concerns how we are to understand the claim that mathematics is indispensable. We address this in the next section. The second question concerns the first premise. It is nowhere near as self-evident as the second and it clearly needs some defense. We’ll discuss its defense in the following section. We’ll then present some of the more important objections to the argument, before considering the Quine-Putnam argument’s role in the larger scheme of things — where it stands in relation to other influential arguments for and against mathematical realism.

The question of how we should understand ‘indispensability’ in the present context is crucial to the Quine-Putnam argument, and yet it has received surprisingly little attention. Quine actually speaks in terms of the entities quantified over in the canonical form of our best scientific theories rather than indispensability. Still, the debate continues in terms of indispensability, so we would be well served to clarify this term.

The first thing to note is that ‘dispensability’ is not the same as ‘eliminability’. If this were not so, every entity would be dispensable (due to a theorem of Craig). [3] What we require for an entity to be ‘dispensable’ is for it to be eliminable and that the theory resulting from the entity’s elimination be an attractive theory. (Perhaps, even stronger, we require that the resulting theory be more attractive than the original.) We will need to spell out what counts as an attractive theory but for this we can appeal to the standard desiderata for good scientific theories: empirical success; unificatory power; simplicity; explanatory power; fertility and so on. Of course there will be debate over what desiderata are appropriate and over their relative weightings, but such issues need to be addressed and resolved independently of issues of indispensability. (See Burgess (1983) and Colyvan (1999) for more on these issues.)

These issues naturally prompt the question of how much mathematics is indispensable (and hence how much mathematics carries ontological commitment). It seems that the indispensability argument only justifies belief in enough mathematics to serve the needs of science. Thus we find Putnam speaking of “the set theoretic ‘needs’ of physics” (Putnam 1979b, p. 346) and Quine claiming that the higher reaches of set theory are “mathematical recreation ... without ontological rights” (Quine 1986, p. 400) since they do not find physical applications. One could take a less restrictive line and claim that the higher reaches of set theory, although without physical applications, do carry ontological commitment by virtue of the fact that they have applications in other parts of mathematics. So long as the chain of applications eventually “bottoms out” in physical science, we could rightfully claim that the whole chain carries ontological commitment. Quine himself justifies some transfinite set theory along these lines (Quine 1984, p. 788), but he sees no reason to go beyond the constructible sets (Quine 1986, p. 400). His reasons for this restriction, however, have little to do with the indispensability argument and so supporters of this argument need not side with Quine on this issue.

Although both premises of the Quine-Putnam indispensability argument have been questioned, it’s the first premise that is most obviously in need of support. This support comes from the doctrines of naturalism and holism.

Following Quine, naturalism is usually taken to be the philosophical doctrine that there is no first philosophy and that the philosophical enterprise is continuous with the scientific enterprise (Quine 1981b). By this Quine means that philosophy is neither prior to nor privileged over science. What is more, science, thus construed (i.e. with philosophy as a continuous part) is taken to be the complete story of the world. This doctrine arises out of a deep respect for scientific methodology and an acknowledgment of the undeniable success of this methodology as a way of answering fundamental questions about all nature of things. As Quine suggests, its source lies in “unregenerate realism, the robust state of mind of the natural scientist who has never felt any qualms beyond the negotiable uncertainties internal to science” (Quine 1981b, p. 72). For the metaphysician this means looking to our best scientific theories to determine what exists, or, perhaps more accurately, what we ought to believe to exist. In short, naturalism rules out unscientific ways of determining what exists. For example, naturalism rules out believing in the transmigration of souls for mystical reasons. Naturalism would not, however, rule out the transmigration of souls if our best scientific theories were to require the truth of this doctrine. [4]

Naturalism, then, gives us a reason for believing in the entities in our best scientific theories and no other entities. Depending on exactly how you conceive of naturalism, it may or may not tell you whether to believe in all the entities of your best scientific theories. We take it that naturalism does give us some reason to believe in all such entities, but that this is defeasible. This is where holism comes to the fore: in particular, confirmational holism.

Confirmational holism is the view that theories are confirmed or disconfirmed as wholes (Quine 1980b, p. 41). So, if a theory is confirmed by empirical findings, the whole theory is confirmed. In particular, whatever mathematics is made use of in the theory is also confirmed (Quine 1976, pp. 120–122). Furthermore, it is the same evidence that is appealed to in justifying belief in the mathematical components of the theory that is appealed to in justifying the empirical portion of the theory (if indeed the empirical can be separated from the mathematical at all). Naturalism and holism taken together then justify P1 . Roughly, naturalism gives us the “only” and holism gives us the “all” in P1.

It is worth noting that in Quine’s writings there are at least two holist themes. The first is the confirmational holism discussed above (often called the Quine-Duhem thesis). The other is semantic holism which is the view that the unit of meaning is not the single sentence, but systems of sentences (and in some extreme cases the whole of language). This latter holism is closely related to Quine’s well-known denial of the analytic-synthetic distinction (Quine 1980b) and his equally famous indeterminacy of translation thesis (Quine 1960). Although for Quine, semantic holism and confirmational holism are closely related, there is good reason to distinguish them, since the former is generally thought to be highly controversial while the latter is considered relatively uncontroversial.

Why this is important to the present debate is that Quine explicitly invokes the controversial semantic holism in support of the indispensability argument (Quine 1980b, pp. 45–46). Most commentators, however, are of the view that only confirmational holism is required to make the indispensability argument fly (see, for example, Colyvan (1998a); Field (1989, pp. 14–20); Hellman (1999); Resnik (1995a; 1997); Maddy (1992)) and my presentation here follows that accepted wisdom. It should be kept in mind, however, that while the argument, thus construed, is Quinean in flavor it is not, strictly speaking, Quine’s argument.

There have been many objections to the indispensability argument, including Charles Parsons’ (1980) concern that the obviousness of basic mathematical statements is left unaccounted for by the Quinean picture and Philip Kitcher’s (1984, pp. 104–105) worry that the indispensability argument doesn’t explain why mathematics is indispensable to science. The objections that have received the most attention, however, are those due to Hartry Field, Penelope Maddy and Elliott Sober. In particular, Field’s nominalisation program has dominated recent discussions of the ontology of mathematics.

Field (2016) presents a case for denying the second premise of the Quine-Putnam argument. That is, he suggests that despite appearances mathematics is not indispensable to science. There are two parts to Field’s project. The first is to argue that mathematical theories don’t have to be true to be useful in applications, they need merely to be conservative. (This is, roughly, that if a mathematical theory is added to a nominalist scientific theory, no nominalist consequences follow that wouldn’t follow from the nominalist scientific theory alone.) This explains why mathematics can be used in science but it does not explain why it is used. The latter is due to the fact that mathematics makes calculation and statement of various theories much simpler. Thus, for Field, the utility of mathematics is merely pragmatic — mathematics is not indispensable after all.

The second part of Field’s program is to demonstrate that our best scientific theories can be suitably nominalised. That is, he attempts to show that we could do without quantification over mathematical entities and that what we would be left with would be reasonably attractive theories. To this end he is content to nominalise a large fragment of Newtonian gravitational theory. Although this is a far cry from showing that all our current best scientific theories can be nominalised, it is certainly not trivial. The hope is that once one sees how the elimination of reference to mathematical entities can be achieved for a typical physical theory, it will seem plausible that the project could be completed for the rest of science. [5]

There has been a great deal of debate over the likelihood of the success of Field’s program but few have doubted its significance. Recently, however, Penelope Maddy, has pointed out that if P1 is false, Field’s project may turn out to be irrelevant to the realism/anti-realism debate in mathematics.

Maddy presents some serious objections to the first premise of the indispensability argument (Maddy 1992; 1995; 1997). In particular, she suggests that we ought not have ontological commitment to all the entities indispensable to our best scientific theories. Her objections draw attention to problems of reconciling naturalism with confirmational holism. In particular, she points out how a holistic view of scientific theories has problems explaining the legitimacy of certain aspects of scientific and mathematical practices. Practices which, presumably, ought to be legitimate given the high regard for scientific practice that naturalism recommends. It is important to appreciate that her objections, for the most part, are concerned with methodological consequences of accepting the Quinean doctrines of naturalism and holism — the doctrines used to support the first premise. The first premise is thus called into question by undermining its support.

Maddy’s first objection to the indispensability argument is that the actual attitudes of working scientists towards the components of well-confirmed theories vary from belief, through tolerance, to outright rejection (Maddy 1992, p. 280). The point is that naturalism counsels us to respect the methods of working scientists, and yet holism is apparently telling us that working scientists ought not have such differential support to the entities in their theories. Maddy suggests that we should side with naturalism and not holism here. Thus we should endorse the attitudes of working scientists who apparently do not believe in all the entities posited by our best theories. We should thus reject P1 .

The next problem follows from the first. Once one rejects the picture of scientific theories as homogeneous units, the question arises whether the mathematical portions of theories fall within the true elements of the confirmed theories or within the idealized elements. Maddy suggests the latter. Her reason for this is that scientists themselves do not seem to take the indispensable application of a mathematical theory to be an indication of the truth of the mathematics in question. For example, the false assumption that water is infinitely deep is often invoked in the analysis of water waves, or the assumption that matter is continuous is commonly made in fluid dynamics (Maddy 1992, pp. 281–282). Such cases indicate that scientists will invoke whatever mathematics is required to get the job done, without regard to the truth of the mathematical theory in question (Maddy 1995, p. 255). Again it seems that confirmational holism is in conflict with actual scientific practice, and hence with naturalism. And again Maddy sides with naturalism. (See also Parsons (1983) for some related worries about Quinean holism.) The point here is that if naturalism counsels us to side with the attitudes of working scientists on such matters, then it seems that we ought not take the indispensability of some mathematical theory in a physical application as an indication of the truth of the mathematical theory. Furthermore, since we have no reason to believe that the mathematical theory in question is true, we have no reason to believe that the entities posited by the (mathematical) theory are real. So once again we ought to reject P1 .

Maddy’s third objection is that it is hard to make sense of what working mathematicians are doing when they try to settle independent questions. These are questions, that are independent of the standard axioms of set theory — the ZFC axioms. [6] In order to settle some of these questions, new axiom candidates have been proposed to supplement ZFC, and arguments have been advanced in support of these candidates. The problem is that the arguments advanced seem to have nothing to do with applications in physical science: they are typically intra-mathematical arguments. According to indispensability theory, however, the new axioms should be assessed on how well they cohere with our current best scientific theories. That is, set theorists should be assessing the new axiom candidates with one eye on the latest developments in physics. Given that set theorists do not do this, confirmational holism again seems to be advocating a revision of standard mathematical practice, and this too, claims Maddy, is at odds with naturalism (Maddy 1992, pp. 286–289).

Although Maddy does not formulate this objection in a way that directly conflicts with P1 it certainly illustrates a tension between naturalism and confirmational holism. [7] And since both these are required to support P1, the objection indirectly casts doubt on P1. Maddy, however, endorses naturalism and so takes the objection to demonstrate that confirmational holism is false. We’ll leave the discussion of the impact the rejection of confirmational holism would have on the indispensability argument until after we outline Sober’s objection, because Sober arrives at much the same conclusion.

Elliott Sober’s objection is closely related to Maddy’s second and third objections. Sober (1993) takes issue with the claim that mathematical theories share the empirical support accrued by our best scientific theories. In essence, he argues that mathematical theories are not being tested in the same way as the clearly empirical theories of science. He points out that hypotheses are confirmed relative to competing hypotheses. Thus if mathematics is confirmed along with our best empirical hypotheses (as indispensability theory claims), there must be mathematics-free competitors. But Sober points out that all scientific theories employ a common mathematical core. Thus, since there are no competing hypotheses, it is a mistake to think that mathematics receives confirmational support from empirical evidence in the way other scientific hypotheses do.

This in itself does not constitute an objection to P1 of the indispensability argument, as Sober is quick to point out (Sober 1993, p. 53), although it does constitute an objection to Quine’s overall view that mathematics is part of empirical science. As with Maddy’s third objection, it gives us some cause to reject confirmational holism. The impact of these objections on P1 depends on how crucial you think confirmational holism is to that premise. Certainly much of the intuitive appeal of P1 is eroded if confirmational holism is rejected. In any case, to subscribe to the conclusion of the indispensability argument in the face of Sober’s or Maddy’s objections is to hold the position that it’s permissible at least to have ontological commitment to entities that receive no empirical support. This, if not outright untenable, is certainly not in the spirit of the original Quine-Putnam argument.

The arguments against holism from Maddy and Sober resulted in a reevaluation of the indispensability argument. If, contra Quine, scientists do not accept all the entities of our best scientific theories, where does this leave us? We need criteria for when to treat posits realistically. Here is where the debate over the indispensability argument took an interesting turn. Scientific realists, at least, accept those posits of our best scientific theories that contribute to scientific explanations. According to this line of thought, we ought to believe in electrons, say, not because they are indispensable to our best scientific theories but because they are indispensable in a very specific way: they are explanatorily indispensable. If mathematics could be shown to contribute to scientific explanations in this way, mathematical realism would again be on par with scientific realism. Indeed, this is the focus of most of the contemporary discussion on the indispensability argument. The central question is: does mathematics contribute to scientific explanations and if so, does it do it in the right kind of way.

One example of how mathematics might be thought to be explanatory is found in the periodic cicada case (Yoshimura 1997 and Baker 2005). North American Magicicadas are found to have life cycles of 13 or 17 years. It is proposed by some biologists that there is an evolutionary advantage in having such prime-numbered life cycles. Prime-numbered life cycles mean that the Magicicadas avoid competition, potential predators, and hybridisation. The idea is quite simple: because prime numbers have no non-trivial factors, there are very few other life cycles that can be synchronised with a prime-numbered life cycle. The Magicicadas thus have an effective avoidance strategy that, under certain conditions, will be selected for. While the explanation being advanced involves biology (e.g. evolutionary theory, theories of competition and predation), a crucial part of the explanation comes from number theory, namely, the fundamental fact about prime numbers. Baker (2005) argues that this is a genuinely mathematical explanation of a biological fact. There are other examples of alleged mathematical explanations in the literature but this remains the most widely discussed and is something of a poster child for mathematical explanation.

Questions about this case focus on whether the mathematics is really contributing to the explanation (or whether it is merely standing in for the biological facts and it is these that really do the explaining), whether the alleged explanation is an explanation at all, and whether the mathematics in question is involved in the explanation in the right kind of way. Finally, it is worth mentioning that although the recent interest in mathematical explanation arose out of debates over the indispensability argument, the status of mathematical explanations in the empirical sciences has also attracted interest in its own right. Moreover, such explanations (sometimes called “extra-mathematical explanations”) lead one very naturally to think about explanations of mathematical facts by appeal to further mathematical facts (sometimes called “intra-mathematical explanation”). These two kinds of mathematical explanation are related, of course. If, for example, some theorem of mathematics has its explanation rest in an explanatory proof, then any applications of that theorem in the empirical realm would give rise to a prima facie case that the full explanation of the empirical phenomenon in question involves the intra-mathematical explanation of the theorem. For these and other reasons, both kinds of mathematical explanation have attracted a great deal of interest from philosophers of mathematics and philosophers of science in recent years.

It is not clear how damaging the above criticisms are to the indispensability argument and whether the explanatory version of the argument survives. Indeed, the debate is very much alive, with many recent articles devoted to the topic. (See bibliography notes below.) Closely related to this debate is the question of whether there are any other decent arguments for platonism. If, as some believe, the indispensability argument is the only argument for platonism worthy of consideration, then if it fails, platonism in the philosophy of mathematics seems bankrupt. Of relevance then is the status of other arguments for and against mathematical realism. In any case, it is worth noting that the indispensability argument is one of a small number of arguments that have dominated discussions of the ontology of mathematics. It is therefore important that this argument not be viewed in isolation.

The two most important arguments against mathematical realism are the epistemological problem for platonism — how do we come by knowledge of causally inert mathematical entities? (Benacerraf 1983b) — and the indeterminacy problem for the reduction of numbers to sets — if numbers are sets, which sets are they (Benacerraf 1983a)? Apart from the indispensability argument, the other major argument for mathematical realism appeals to a desire for a uniform semantics for all discourse: mathematical and non-mathematical alike (Benacerraf 1983b). Mathematical realism, of course, meets this challenge easily, since it explains the truth of mathematical statements in exactly the same way as in other domains. [8] It is not so clear, however, how nominalism can provide a uniform semantics.

Finally, it is worth stressing that even if the indispensability argument is the only good argument for platonism, the failure of this argument does not necessarily authorize nominalism, for the latter too may be without support. It does seem fair to say, however, that if the objections to the indispensability argument are sustained then one of the most important arguments for platonism is undermined. This would leave platonism on rather shaky ground.

Although the indispensability argument is to be found in many places in Quine’s writings (including 1976; 1980a; 1980b; 1981a; 1981c), the locus classicus is Putnam’s short monograph Philosophy of Logic (included as a chapter of the second edition of the third volume of his collected papers (Putnam, 1979b)). See also Putnam (1979a) and the introduction of Field (1989), which has an excellent outline of the argument. Colyvan (2001) presents a sustained defence of the argument.

See Chihara (1973), and Field (1989; 2016) for attacks on the second premise and Colyvan (1999; 2001), Lyon and Colyvan (2008), Maddy (1990), Malament (1982), Resnik (1985), Shapiro (1983) and Urquhart (1990) for criticisms of Field’s program. See the preface to the second edition of Field 2016 for a good retrospective on these debates. For a fairly comprehensive look at nominalist strategies in the philosophy of mathematics (including an excellent discussion of Field’s program), see Burgess and Rosen (1997), while Feferman (1993) questions the amount of mathematics required for empirical science. See Azzouni (1997; 2004; 2012), Balaguer (1996b; 1998), Bueno (2012), Leng (2002; 2010; 2012), Liggins (2012), Maddy (1992; 1995; 1997), Melia (2000; 2002), Peressini (1997), Pincock (2004), Sober (1993), Vineberg (1996) and Yablo (1998; 2005; 2012) for attacks on the first premise. Baker (2001; 2005; 2012), Bangu (2012), Colyvan (1998a; 2001; 2002; 2007; 2010; 2012), Hellman (1999) and Resnik (1995a; 1997) reply to some of these objections.

For variants of the Quinean indispensability argument see Maddy (1992) and Resnik (1995a).

There has been a great deal of recent literature on the explanatory version of the indispensability argument. Early presentations of such an argument can be found in Colyvan (1998b; 2002), and most explicitly in Baker (2005), although this work was anticipated by Steiner (1978a; 1978b) on mathematical explanation and Smart on geometric explanation (1990). Some of the key articles on the explanatory version of the argument include Baker (2005; 2009; 2012; 2017; 2021), Bangu (2008; 2013), Baron (2014), Batterman (2010), Bueno and French (2012), Colyvan (2002; 2010; 2012; 2018), Lyon (2012), Rizza (2011), Saatsi (2011; 2016) and Yablo (2012).

Arising out of this debate over the role of mathematical explanation in indispensability arguments, has been a renewed interest in mathematical explanation for its own sake. This includes work on reconciling mathematical explanations in science with other forms of scientific explanation as well as investigating explanation within mathematics itself. Some of this work includes: Baron (2016), Baron et al. (2017; 2020), Colyvan et al. (2018), Lange (2017), Mancosu (2008), and Pincock (2011).

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abduction | mathematical: explanation | meaning holism | naturalism | nominalism: in metaphysics | Platonism: in metaphysics | Quine, Willard Van Orman | realism

## Acknowledgments

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## Indispensability of Mathematics by Sorin Bangu LAST REVIEWED: 10 November 2022 LAST MODIFIED: 27 March 2014 DOI: 10.1093/obo/9780195396577-0241

Philosophers speak of the “indispensability of mathematics” in association with a family of arguments, taking as their starting point two claims. First, a descriptive claim: science and everyday discourse is permeated by mathematics and mathematical terms. Second, a more controversial normative claim: mathematics enjoys a special status within our body of beliefs, because it plays an essential, indeed indispensable, role in our understanding of the world. Although the descriptive thesis is usually regarded as unproblematic, the normative thesis has attracted a good deal of attention. Most importantly, it gave rise to an important metaphysical project, the proposal of a novel argument for an old idea, mathematical realism . The very possibility that such an argument might be successful has led many to reconsider the viability of a realist metaphysics of mathematics traceable to Plato (“mathematical Platonism”), according to which mathematical objects (numbers, sets, etc.) genuinely exist (as opposed to being mere useful fictions), and mathematical truths are objective (as opposed to expressing mere conventions). Several versions of the indispensability argument have been proposed in order to support several forms of realism, and all of them have been subject to sustained criticism, from various angles (although, interestingly, not all objectors are anti-realists, or “nominalists”). The debate around the plausibility of such arguments is connected in subtle ways with other doctrines and positions in the philosophy of science, metaphysics, and epistemology. The work done in this area can be divided into three periods: the pre-Quinean attempt to take the indispensability idea seriously (most notably by Gottlob Frege); the period consisting of the first attempts to articulate such an argument (presented by Willard V. O. Quine and Hilary Putnam, roughly localizable in the period between the 1950s and the 1970s); and finally the “post-Quinean” period (the 1980s to the early 21st century), in which this family of arguments became a relatively well-defined object of philosophical exploration.

## General Overviews

General overviews of the indispensability argument fall into two categories. Some present only the essentials, whereas others go into much more detail and articulate the position more fully, for instance, by examining the possible objections or by discussing the indispensability strategy in relation to other philosophical concerns.

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## Why Is Math Important? 9 Reasons Why Math Skills Improve Quality of Life

Written by Ashley Crowe

- Parent Resources

## Why is math so important in life?

- 9 Benefits of a great math education

## Why students struggle to master certain math concepts

- Set your child’s math skills up for life with Prodigy Math

Math isn't just an important subject in school — it’s essential for many of your daily tasks. You likely use it every day to perform real-life skills, like grocery shopping, cooking and tracking your finances.

What makes math special is that it’s a universal language — a powerful tool with the same meaning across the globe. Though languages divide our world, numbers unite us. Math allows us to work together towards new innovations and ideas.

In this post, learn why math is important for kids and adults. Plus, find out why learning even the most basic math can significantly improve your family’s quality of life.

You simply can’t make it through a day without using some sort of basic math. Here’s why.

A person needs an understanding of math, measurements and fractions to cook and bake. Many people may also use math to count calories or nutrients as part of their diet or exercise routine.

You also need math to calculate when you should leave your house to arrive on time, or how much paint you need to redo your bedroom walls.

And then the big one, money. Financial literacy is an incredibly important skill for adults to master. It can help you budget, save and even help you make big decisions like changing careers or buying a home.

Mathematical knowledge may even be connected to many other not-so-obvious benefits. A strong foundation in math can translate into increased understanding and regulation of your emotions, improved memory and better problem-solving skills.

## The importance of math: 9 benefits of a great math education

Math offers more opportunities beyond grade school, middle school and high school. Its applications to real-life scenarios are vast.

Though many students sit in math class wondering when they’ll ever use these things they’re learning, we know there are many times their math skills will be needed in adulthood.

The importance of mathematics to your child’s success can’t be overstated. Basic math is a necessity, but even abstract math can help hone critical thinking skills — even if your child chooses not to pursue a STEM-style career. Math can help them succeed professionally, emotionally and cognitively. Here’s why.

## 1. Math promotes healthy brain function

“Use it or lose it.” We hear this said about many skills, and math is no exception.

Solving math problems and improving our math skills gives our brain a good workout. And it improves our cognitive skills over time. Many studies have shown that routinely practicing math keeps our brain healthy and functioning well.

## 2. Math improves problem-solving skills

At first, classic math problems like Johnny bringing home 42 watermelons and returning 13 of them can just seem a silly exercise. But all those math word problems our children solve really do improve their problem solving skills. Word problems teach kids how to pull out the important information and then manipulate it to find a solution.

Later on, complex life problems take the place of workbooks, but problem-solving still happens the same way. When students understand algorithms and problems more deeply, they can decode the facts and more easily solve the issue. Real-life solutions are found with math and logic.

## 3. Math supports logical reasoning and analytical thinking

A strong understanding of math concepts means more than just number sense. It helps us see the pathways to a solution. Equations and word problems need to be examined before determining the best method for solving them. And in many cases, there’s more than one way to get to the right answer.

It’s no surprise that logical reasoning and analytical thinking improve alongside math skills. Logic skills are necessary at all levels of mathematical education.

## 4. Math develops flexible thinking and creativity

Practicing math has been shown to improve investigative skills, resourcefulness and creativity.

This is because math problems often require us to bend our thinking and approach problems in more than one way. The first process we try might not work. We need flexibility and creativity to think of new pathways to the solution. And just like anything else, this way of thinking is strengthened with practice.

## 5. Math opens up many different career paths

There are many careers that use a large number of math concepts. These include architects, accountants, and scientists.

But many other professionals use math skills every day to complete their jobs. CEOs use math to analyze financials. Mailmen use it to calculate how long it will take them to walk their new route. Graphic designers use math to figure out the appropriate scale and proportions in their designs.

No matter what career path your child chooses, math skills will be beneficial.

## Math skills might become even more important for today's kids!

Math can certainly open up a lot of opportunities for many of us. But did you know that careers which heavily use math are going to be among the fastest-growing jobs by the time kids today start their careers? These jobs include:

- Statisticians
- Data scientists
- Software developers
- Cybersecurity analysts

It's not just STEM jobs that will require math either. Other popular, high-growth careers like nursing and teaching now ask for a minimum knowledge of college-level math.

## 6. Math may boost emotional health

While this research is still in its early days, what we have seen is promising.

The parts of the brain used to solve math problems seem to work together with the parts of the brain that regulate emotions. This suggests that math practice can actually help us cope with difficult situations. In these studies, the better someone was with numerical calculations, the better they were at regulating fear and anger. Strong math skills may even be able to help treat anxiety and depression.

## 7. Math improves financial literacy

Though kids may not be managing their finances now, there's going to be plenty of times where math skills are going to make a massive difference in their life as an adult.

Budgeting and saving is a big one. Where can they cut back on their spending? How will budgeting help them reach their financial goals? Can they afford this new purchase now?

As they age into adulthood, It will benefit your child to understand how loans and interest work before purchasing a house or car. They should fully grasp profits and losses before investing in the stock market. And they will likely need to evaluate job salaries and benefits before choosing their first job.

## 8. Math sharpens your memory

Learning mental math starts in elementary school. Students learn addition tables, then subtraction, multiplication and division tables. As they master those skills, they’ll begin to memorize more tips and tricks, like adding a zero to the end when multiplying by 10. Students will memorize algorithms and processes throughout their education.

Using your memory often keeps it sharp. As your child grows and continues to use math skills in adulthood, their memory will remain in tip top shape.

## 9. Math teaches perseverance

“I can do it!’

These are words heard often from our toddlers. This phrase is a marker of growth, and a point of pride. But as your child moves into elementary school, you may not hear these words as often or with as much confidence as before.

Learning math is great for teaching perseverance. With the right math instruction, your child can see their progress and once again feel that “I can do it” attitude. The rush of excitement a child experiences when they master a new concept sticks in their memory. And they can reflect back on it when they’re struggling with a new, harder skill.

Even when things get tough, they’ll know they can keep trying and eventually overcome it — because they’ve done it before.

## Tip: Set goals to inspire and motivate your child to learn math

If your child has a Prodigy Math Membership , you can use your parent account to set learning goals for them to achieve as they play our online math game.

The best bit? Every time they complete a goal, they'll also get a special in-game reward!

Many students experience roadblocks and hurdles throughout their math education. You might recognize some of these math struggles below in your child. But don’t worry! Any struggle is manageable with the right support and help. Together, you and your child can tackle anything.

Here are some of the most common math struggles.

- Increasing complexity

Sometimes the pace of class moves a bit faster than your child can keep up with. Or the concepts are just too abstract and difficult for them to wrap their mind around in one lesson. Some math ideas simply take more time to learn.

- Wrong teaching style

A good teaching style with plenty of practice is essential to a high-quality math education. If the teacher’s style doesn’t mesh well with how your child learns, math class can be challenging.

- Fear of failure

Even as adults, we can feel scared to fail. It’s no surprise that our children experience this same same fear, especially with the many other pressures school can bring.

- Lack of practice

Sometimes, all your child needs is a little more practice. But this can be easier said than done. You can help by providing them with plenty of support and encouragement to help them get that practice time in.

- Math anxiety

Algorithms and complex problems can strike anxiety in the heart of any child (and many adults). Math anxiety is a common phenomenon. But with the right coping strategies it can be managed.

## Set your child’s math skills up for success with Prodigy Math

Now we've discovered just how important math is in both our everyday and life decisions, let's set the next generation up for success with the right tools that'll help them learn math.

Prodigy Math is a game-based, online learning platform that makes learning math fun for kids. As kids play and explore a safe, virtual world filled with fun characters and pets to collect, they'll answer math questions. These questions are curriculum-aligned and powered by an adaptive algorithm that can help them master math skills more quickly.

Plus, with a free parent account , you'll also get to be a big part of their math education without needing to be a math genius. You'll get to:

- Easily keep up with their math learning with a monthly Report Card
- See how they're doing in math class when their teacher uses Prodigy Math
- Send them motivational messages to encourage their perseverance in math

Want to play an even bigger role in helping your child master math? Try our optional Math Memberships for extra in-game content for your child to enjoy and get amazing parent tools like the ability to set in-game goals and rewards for them to achieve.

See why Prodigy can make math fun below!

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## Why is Math Important: Benefits of Learning Math at School

Created: January 11, 2024

Last updated: January 11, 2024

Why is math important is a question worth exploring. Mathematics, a subject beyond mere numbers and formulas, constitutes the core of our existence. Its influence extends far beyond the confines of textbooks, penetrating the very essence of modern life. The topic — why is math the most important subject — also carries weight within the realm of education, which is why kids may be asked to write a why is math important essay in class.

As we embrace math in education, we enable ourselves to unravel the mysteries of our reality. Through this article, you will discover answers to the question, why is math so important, and understand the many benefits of immersing ourselves in mathematics.

## Why Is Math Important for Kids to Learn?

Math plays a significant role in everyone’s educational journey, bringing many benefits beyond just numbers. From the basics like counting and recognizing shapes to more complicated aspects like algebra, geometry, and calculus, studying math grounds students intellectually. At its heart, math teaches discipline and accuracy.

As people study math, they learn to take logical steps, follow the rules, and pay attention to the details. These skills make their studies easier and help them in other areas of life, teaching them how to approach problems systematically. Math also hones critical thinking and analysis.

It’s essential to know the answer to the question — why is math important for kids. When faced with math problems, we learn to spot patterns, make connections, and develop hypotheses. This natural problem-solving pathway helps us understand how things work and resolve complex issues. Besides, math literacy is a must-have in a world full of data and tech. Knowing the ins and outs of math gives kids the ability to interpret numbers and make well-thought-out decisions in terms of finance, health, and science.

Also, math takes students and even teachers to the apex of creativity! When both parties explore numbers, shapes, and equations, they use their imaginations and develop new ways to solve problems and develop ideas. Finally, math encourages collaboration. Group activities and conversations about math help them communicate better, learn together, and make friends.

Having understood the overview of math’s relevance in people’s lives, let’s delve deeper into why is math important in everyday life for kids.

## Math Hones Complex Problem-Solving Skills

Knowing the answer to the ‘why is math important in life’ question enables kids to break down complex problems into smaller components, identify pertinent variables, and use appropriate formulas or methods to arrive at practical solutions. Math equips kids with a structured approach to problem-solving, empowering them to overcome obstacles and adapt to a dynamic world.

The capacity to methodically resolve issues enables them to approach various challenges with unwavering confidence and creativity, whether resolving complex technical troubleshooting issues, streamlining workflows, or interpersonal conflicts.

## It Promotes Critical Thinking

Knowing 5 reasons why math is important reveals math’s role in fostering critical thinking. The journey of solving mathematical problems is crucible for developing critical thinking. As kids immerse themselves in scrutinizing data, solving maze-like word problems, and developing logical strategies, they develop a robust skill — evaluating information from diverse perspectives.

This ability to see recurring patterns and coherent conclusions is essential to making informed decisions. In debate and dialogue, kids with sharp critical thinking demonstrate the ability to obtain reliable sources, deconstruct complex arguments, and participate meaningfully in discussions. When faced with unexpected circumstances or a whirlwind of rapidly changing scenarios, this honed analytical skill allows them to objectively weigh new information, seamlessly adjust strategies, and deftly navigate the tides of change.

## Math Improves Kids Financial Knowledge

Why math is the most important subject is validated within academics, but we can look beyond that. Math knowledge is a vital component of financial literacy, as it provides kids with the understanding and tools to make informed decisions that shape their financial well-being. Mathematics is central in helping individuals develop the essential skills to decipher complex financial concepts.

From understanding the dynamics of interest rates and the complex effects of investing to evaluating risk and return profiles, mathematics provides the basis for building a solid financial foundation. Using these mathematical insights, kids can create an adequate budget that meets their goals and desires. But financial literacy goes beyond self-interest; it enables them to contribute positively to their communities.

By making intelligent philanthropic decisions or supporting local businesses, financially savvy kids become agents of change that drive economic growth and community development. Financial literacy provides clarity about a student’s perspective on broader financial issues.

Using mathematical reasoning, they can engage in informed discussions about public policy, evaluate economic proposals, and make informed choices with far-reaching societies. The combination of mathematics and financial literacy, allowing them to secure their financial future and actively participate in creating a more financially stable and fair society, makes us more confident in answering the question, why is math important?

## It Helps Kids Develop Technical Skills

In a digital age where technology permeates every aspect of modern life, the question of ‘why is discrete math important’ is quickly answered. Look at cybersecurity, for example. In 2023, it is among the most sought-after technical skills as companies try to protect their networks and data from breaches and uphold customers’ privacy.

A good understanding of mathematics opens up the ability to understand, analyze and innovate in a complex digital environment. Knowledge of mathematics allows kids to contribute to the development of technology actively.

As technology evolves and shapes the future, mathematicians are uniquely positioned to drive progress. Using mathematical principles, they confidently explore the digital world, contributing to developing new solutions, advanced applications, and transformative breakthroughs that move society into uncharted territories of technological innovation.

## Math Opens The Door to More Career Opportunities

Kids know why math is important and impacts job opportunities because of how many more career paths it offers them. Beyond the bounds of traditional math-oriented roles like engineering and finance, the need for math skills has permeated many industries. Meanwhile, dynamic marketing has used statistical analysis to discover consumer behavior, improve customer segmentation, and drive strategic campaigns.

As artificial intelligence and automation redefine industries, kids with a solid foundation in mathematics have the adaptability and innovation to thrive in new areas of employment in the future. From harnessing the power of big data to building data-driven narratives, these math-savvy professionals are at the forefront of shaping the future of work.

## Learning Math Improves Analytical Skills

Mathematical analysis is crucial for developing analytical thinking, an invaluable skill in our complex, information-saturated world. So why is it important to learn math to improve analytical skills? In an age where navigating massive data sets and deciphering multifaceted challenges is the norm, the ability to discern complex situations and evaluate evidence becomes valuable.

In a world where career paths and problem-solving paradigms are evolving at an unprecedented rate, the enrichment provided by mathematical and analytical ability is a cornerstone of success. Whether driving an industry into the future or developing innovative solutions to global problems, kids with these skills are built to make a lasting and transformative impact.

## Progressive Scientific Discovery

The question — why is math and science important — is a run-off of the belief that math is often the language of science. Math is an indispensable tool for pushing the boundaries of scientific research and inquiry. It is the hidden force behind the breakthrough discoveries that allow scientists to bridge the gap between theoretical concepts and empirical observations.

Clinical trial design and medical analysis are governed by mathematical principles, which aid researchers in evaluating the efficacy of interventions and treatments. Statistical methods rooted in mathematics can provide insight into the effects of new drugs, the spread of diseases, and the impact of public health initiatives. This quantitative approach that improves medical knowledge and saves lives by guiding evidence-based medical practice shows why learning math is important.

## It Helps Kids Develop Mental Stamina and Endurance

Facing complex math problems develops mathematical and endurance skills. It promotes mental strength and a will to overcome difficulties. When kids become aware of this, before you point out 5 reasons why math is not important they can already give you countless reasons why math is important. That is because they have gone through the rigors of solving math and now understand that mastery requires dedication, persistence, and a willingness to face failure.

They develop an inherent resilience beyond mathematics as they solve complex problems and grapple with confusing concepts. This little thing becomes the foundation of personal and professional success. Those who successfully navigate the difficulties of mathematics are better prepared to face the complexity of the modern world.

As we ponder why math is important in life, we should know that math provides a compass for navigating complexity in a world of information and rapid developments. This article could have still gone ahead to give an extra 10 reasons why math is important as its importance is countless. But you get the point already!

Mastering mathematics nurtures critical thinking, problem-solving abilities, and analytical reasoning — qualities necessary in a world filled with complex challenges and diverse opportunities. These all make us understand why math is important in our daily lives.

But why is learning math important at Brighterly? Brighterly recognizes the transformative power of mathematics and its role in shaping resilient individuals. They provided a platform that supports math understanding and learning. So register now to embark on a journey of discovery, where interactive lessons, engaging activities, and a supportive community await.

Jessica is a a seasoned math tutor with over a decade of experience in the field. With a BSc and Master’s degree in Mathematics, she enjoys nurturing math geniuses, regardless of their age, grade, and skills. Apart from tutoring, Jessica blogs at Brighterly. She also has experience in child psychology, homeschooling and curriculum consultation for schools and EdTech websites.

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## What Students Are Saying About the Value of Math

We asked teenagers: Do you see the point in learning math? The answer from many was “yes.”

By The Learning Network

“Mathematics, I now see, is important because it expands the world,” Alec Wilkinson writes in a recent guest essay . “It is a point of entry into larger concerns. It teaches reverence. It insists one be receptive to wonder. It requires that a person pay close attention.”

In our writing prompt “ Do You See the Point in Learning Math? ” we wanted to know if students agreed. Basic arithmetic, sure, but is there value in learning higher-level math, such as algebra, geometry and calculus? Do we appreciate math enough?

The answer from many students — those who love and those who “detest” the subject alike — was yes. Of course math helps us balance checkbooks and work up budgets, they said, but it also helps us learn how to follow a formula, appreciate music, draw, shoot three-pointers and even skateboard. It gives us different perspectives, helps us organize our chaotic thoughts, makes us more creative, and shows us how to think rationally.

Not all were convinced that young people should have to take higher-level math classes all through high school, but, as one student said, “I can see myself understanding even more how important it is and appreciating it more as I get older.”

Thank you to all the teenagers who joined the conversation on our writing prompts this week, including students from Bentonville West High School in Centerton, Ark, ; Harvard-Westlake School in Los Angeles ; and North High School in North St. Paul, Minn.

Please note: Student comments have been lightly edited for length, but otherwise appear as they were originally submitted.

## “Math is a valuable tool and function of the world.”

As a musician, math is intrinsically related to my passion. As a sailor, math is intertwined with the workings of my boat. As a human, math is the building block for all that functions. When I was a child, I could very much relate to wanting a reason behind math. I soon learned that math IS the reason behind all of the world’s workings. Besides the benefits that math provides to one’s intellect, it becomes obvious later in life that math is a valuable tool and function of the world. In music for example, “adolescent mathematics” are used to portray functions of audio engineering. For example, phase shifting a sine wave to better project sound or understanding waves emitted by electricity and how they affect audio signals. To better understand music, math is a recurring pattern of intervals between generating pitches that are all mathematically related. The frets on a guitar are measured precisely to provide intervals based on a tuning system surrounding 440Hz, which is the mathematically calculated middle of the pitches humans can perceive and a string can effectively generate. The difference between intervals in making a chord are not all uniform, so guitar frets are placed in a way where all chords can sound equally consonant and not favor any chord. The power of mathematics! I am fascinated by the way that math creeps its way into all that I do, despite my plentiful efforts to keep it at a safe distance …

— Renan, Miami Country Day School

## “Math isn’t about taking derivatives or solving for x, it’s about having the skills to do so and putting them to use elsewhere in life.”

I believe learning mathematics is both crucial to the learning and development of 21st century students and yet also not to be imposed upon learners too heavily. Aside from the rise in career opportunity in fields centered around mathematics, the skills gained while learning math are able to be translated to many facets of life after a student’s education. Learning mathematics develops problem solving skills which combine logic and reasoning in students as they grow. The average calculus student may complain of learning how to take derivatives, arguing that they will never have to use this after high school, and in that, they may be right. Many students in these math classes will become writers, musicians, or historians and may never take a derivative in their life after high school, and thus deem the skill to do so useless. However, learning mathematics isn’t about taking derivatives or solving for x, it’s about having the skills to do so and putting them to use elsewhere in life. A student who excels at calculus may never use it again, but with the skills of creativity and rational thinking presented by this course, learning mathematics will have had a profound effect on their life.

— Cam, Glenbard West

## “Just stop and consider your hobbies and pastimes … all of it needs math.”

Math is timing, it’s logic, it’s precision, it’s structure, and it’s the way most of the physical world works. I love math — especially algebra and geometry — as it all follows a formula, and if you set it up just right, you can create almost anything you want in at least two different ways. Just stop and consider your hobbies and pastimes. You could be into skateboarding, basketball, or skiing. You could be like me, and sit at home for hours on end grinding out solves on a Rubik’s cube. Or you could be into sketching. Did you know that a proper drawing of the human face places the eyes exactly halfway down from the top of the head? All of it needs math. Author Alec Wilkinson, when sharing his high school doubting view on mathematics, laments “If I had understood how deeply mathematics is embedded in the world …” You can’t draw a face without proportions. You can’t stop with your skis at just any angle. You can’t get three points without shooting at least 22 feet away from the basket, and get this: you can’t even ride a skateboard if you can’t create four congruent wheels to put on it.

— Marshall, Union High School, Vancouver, WA

## “Math gives us a different perspective on everyday activities.”

Even though the question “why do we even do math?” is asked all the time, there is a deeper meaning to the values it shares. Math gives us a different perspective on everyday activities, even if those activities in our routine have absolutely nothing to do with mathematical concepts itself. Geometry, for instance, allows us to think on a different level than simply achieving accuracy maintains. It trains our mind to look at something from various viewpoints as well as teaching us to think before acting and organizing chaotic thoughts. The build up of learning math can allow someone to mature beyond the point where if they didn’t learn math and thought through everything. It paves a way where we develop certain characteristics and traits that are favorable when assisting someone with difficult tasks in the future.

— Linden, Harvard-Westlake High School, CA

## “Math teaches us how to think.”

As explained in the article, math is all around us. Shapes, numbers, statistics, you can find math in almost anything and everything. But is it important for all students to learn? I would say so. Math in elementary school years is very important because it teaches how to do simple calculations that can be used in your everyday life; however middle and high school math isn’t used as directly. Math teaches us how to think. It’s far different from any other subject in school, and truly understanding it can be very rewarding. There are also many career paths that are based around math, such as engineering, statistics, or computer programming, for example. These careers are all crucial for society to function, and many pay well. Without a solid background in math, these careers wouldn’t be possible. While math is a very important subject, I also feel it should become optional at some point, perhaps part way through high school. Upper level math classes often lose their educational value if the student isn’t genuinely interested in learning it. I would encourage all students to learn math, but not require it.

— Grey, Cary High School

## “Math is a valuable tool for everyone to learn, but students need better influences to show them why it’s useful.”

Although I loved math as a kid, as I got older it felt more like a chore; all the kids would say “when am I ever going to use this in real life?” and even I, who had loved math, couldn’t figure out how it benefits me either. This was until I started asking my dad for help with my homework. He would go on and on about how he used the math I was learning everyday at work and even started giving me examples of when and where I could use it, which changed my perspective completely. Ultimately, I believe that math is a valuable tool for everyone to learn, but students need better influences to show them why it’s useful and where they can use it outside of class.

— Lilly, Union High School

## “At the roots of math, it teaches people how to follow a process.”

I do believe that the math outside of arithmetic, percentages, and fractions are the only math skills truly needed for everyone, with all other concepts being only used for certain careers. However, at the same time, I can’t help but want to still learn it. I believe that at the roots of math, it teaches people how to follow a process. All mathematics is about following a formula and then getting the result of it as accurately as possible. It teaches us that in order to get the results needed, all the work must be put and no shortcuts or guesses can be made. Every equation, number, and symbol in math all interconnect with each other, to create formulas that if followed correctly gives us the answer needed. Everything is essential to getting the results needed, and skipping a step will lead to a wrong answer. Although I do understand why many would see no reason to learn math outside of arithmetic, I also see lessons of work ethics and understanding the process that can be applied to many real world scenarios.

— Takuma, Irvine High School

## “I see now that math not only works through logic but also creativity.”

A story that will never finish resembling the universe constantly expanding, this is what math is. I detest math, but I love a never-ending tale of mystery and suspense. If we were to see math as an adventure it would make it more enjoyable. I have often had a closed mindset on math, however, viewing it from this perspective, I find it much more appealing. Teachers urge students to try on math and though it seems daunting and useless, once you get to higher math it is still important. I see now that math not only works through logic but also creativity and as the author emphasizes, it is “a fundamental part of the world’s design.” This view on math will help students succeed and have a more open mindset toward math. How is this never-ending story of suspense going to affect YOU?

— Audrey, Vancouver, WA union high school

## “In some word problems, I encounter problems that thoroughly interest me.”

I believe math is a crucial thing to learn as you grow up. Math is easily my favorite subject and I wish more people would share my enthusiasm. As Alec Wilkinson writes, “Mathematics, I now see, is important because it expands the world.” I have always enjoyed math, but until the past year, I have not seen a point in higher-level math. In some of the word problems I deal with in these classes, I encounter problems that thoroughly interest me. The problems that I am working on in math involve the speed of a plane being affected by wind. I know this is not riveting to everyone, but I thoroughly wonder about things like this on a daily basis. The type of math used in the plane problems is similar to what Alec is learning — trigonometry. It may not serve the most use to me now, but I believe a thorough understanding of the world is a big part of living a meaningful life.

— Rehan, Cary High School

## “Without high school classes, fewer people get that spark of wonder about math.”

I think that math should be required through high school because math is a use-it-or-lose-it subject. If we stop teaching math in high school and just teach it up to middle school, not only will many people lose their ability to do basic math, but we will have fewer and fewer people get that spark of wonder about math that the author had when taking math for a second time; after having that spark myself, I realized that people start getting the spark once they are in harder math classes. At first, I thought that if math stopped being required in high school, and was offered as an elective, then only people with the spark would continue with it, and everything would be okay. After thinking about the consequences of the idea, I realized that technology requires knowing the seemingly unneeded math. There is already a shortage of IT professionals, and stopping math earlier will only worsen that shortage. Math is tricky. If you try your best to understand it, it isn’t too hard. However, the problem is people had bad math teachers when they were younger, which made them hate math. I have learned that the key to learning math is to have an open mind.

— Andrew, Cary High School

## “I think math is a waste of my time because I don’t think I will ever get it.”

In the article Mr. Wilkinson writes, “When I thought about mathematics at all as a boy it was to speculate about why I was being made to learn it, since it seemed plainly obvious that there was no need for it in adult life.” His experience as a boy resonates with my experience now. I feel like math is extremely difficult at some points and it is not my strongest subject. Whenever I am having a hard time with something I get a little upset with myself because I feel like I need to get everything perfect. So therefore, I think it is a waste of my time because I don’t think I will ever get it. At the age of 65 Mr. Wilkinson decided to see if he could learn more/relearn algebra, geometry and calculus and I can’t imagine myself doing this but I can see myself understanding even more how important it is and appreciating it more as I get older. When my dad was young he hated history but, as he got older he learned to appreciate it and see how we can learn from our past mistakes and he now loves learning new things about history.

— Kate, Cary High School

## “Not all children need to learn higher level math.”

The higher levels of math like calculus, algebra, and geometry have shaped the world we live in today. Just designing a house relates to math. To be in many professions you have to know algebra, geometry, and calculus such as being an economist, engineer, and architect. Although higher-level math isn’t useful to some people. If you want to do something that pertains to math, you should be able to do so and learn those high levels of math. Many things children learn in math they will never use again, so learning those skills isn’t very helpful … Children went through so much stress and anxiety to learn these skills that they will never see again in their lives. In school, children are using their time learning calculus when they could be learning something more meaningful that can prepare them for life.

— Julyssa, Hanover Horton High School

## “Once you understand the basics, more math classes should be a choice.”

I believe that once you get to the point where you have a great understanding of the basics of math, you should be able to take more useful classes that will prepare you for the future better, rather than memorizing equations after equations about weird shapes that will be irrelevant to anything in my future. Yes, all math levels can be useful to others’ futures depending on what career path they choose, but for the ones like me who know they are not planning on encountering extremely high level math equations on the daily, we should not have to take math after a certain point.

— Tessa, Glenbard West High School

## “Math could shape the world if it were taught differently.”

If we learned how to balance checkbooks and learn about actual life situations, math could be more helpful. Instead of learning about rare situations that probably won’t come up in our lives, we should be learning how to live on a budget and succeed money-wise. Since it is a required class, learning this would save more people from going into debt and overspending. In schools today, we have to take a specific class that doesn’t sound appealing to the average teenager to learn how to save and spend money responsibly. If it was required in math to learn about that instead of how far Sally has to walk then we would be a more successful nation as a whole. Math could shape the world differently but the way it is taught in schools does not have much impact on everyday life.

— Becca, Bentonville West High School

## “To be honest, I don’t see the point in learning all of the complicated math.”

In a realistic point of view, I need to know how to cut a cake or a piece of pie or know how to divide 25,000 dollars into 10 paychecks. On the other hand, I don’t need to know the arc and angle. I need to throw a piece of paper into a trash can. I say this because, in all reality and I know a lot of people say this but it’s true, when are we actually going to need this in our real world lives? Learning complicated math is a waste of precious learning time unless you desire to have a career that requires these studies like becoming an engineer, or a math professor. I think that the fact that schools are still requiring us to learn these types of mathematics is just ignorance from the past generations. I believe that if we have the technology to complete these problems in a few seconds then we should use this technology, but the past generations are salty because they didn’t have these resources so they want to do the same thing they did when they were learning math. So to be honest, I don’t see the point in learning all of the complicated math but I do think it’s necessary to know the basic math.

— Shai, Julia R Masterman, Philadelphia, PA

Learn more about Current Events Conversation here and find all of our posts in this column .

## Essays About Math: Top 10 Examples and Writing Prompts

Love it or hate it, an understanding of math is said to be crucial to success. So, if you are writing essays about math, read our top essay examples.

Mathematics is the study of numbers, shapes, and space using reason and usually a special system of symbols and rules for organizing them . It can be used for a variety of purposes, from calculating a business’s profit to estimating the mass of a black hole. However, it can be considered “controversial” to an extent.

Most students adore math or regard it as their least favorite. No other core subject has the same infamy as math for generating passionate reactions both for and against it. It has applications in every field, whether basic operations or complex calculus problems. Knowing the basics of math is necessary to do any work properly.

If you are writing essays about Math, we have compiled some essay examples for you to get started.

## 1. Mathematics: Problem Solving and Ideal Math Classroom by Darlene Gregory

2. math essay by prasanna, 3. short essay on the importance of mathematics by jay prakash.

- 4. Math Anxiety by Elias Wong

## 5. Why Math Isn’t as Useless as We Think by Murtaza Ali

1. mathematics – do you love or hate it, 2. why do many people despise math, 3. how does math prepare you for the future, 4. is mathematics an essential skill, 5. mathematics in the modern world.

“The trait of the teacher that is being strict is we know that will really help the students to change. But it will give a stress and pressure to students and that is one of the causes why students begin to dislike math. As a student I want a teacher that is not so much strict and giving considerations to his students. A teacher that is not giving loads of things to do and must know how to understand the reasons of his students.”

Gregory discusses the reasons for most students’ hatred of math and how teachers handle the subject in class. She says that math teachers do not explain the topics well, give too much work, and demand nothing less than perfection. To her, the ideal math class would involve teachers being more considerate and giving less work.

You might also be interested in our ordinal number explainer.

“Math is complicated to learn, and one needs to focus and concentrate more. Math is logical sometimes, and the logic needs to be derived out. Maths make our life easier and more straightforward. Math is considered to be challenging because it consists of many formulas that have to be learned, and many symbols and each symbol generally has its significance.”

In her essay, Prasanna gives readers a basic idea of what math is and its importance. She additionally lists down some of the many uses of mathematics in different career paths, namely managing finances, cooking, home modeling and construction, and traveling. Math may seem “useless” and “annoying” to many, but the essay gives readers a clear message: we need math to succeed.

“In this modern age of Science and Technology, emphasis is given on Science such as Physics, Chemistry, Biology, Medicine and Engineering. Mathematics, which is a Science by any criterion, also is an efficient and necessary tool being employed by all these Sciences. As a matter of fact, all these Sciences progress only with the aid of Mathematics. So it is aptly remarked, ‘Mathematics is a Science of all Sciences and art of all arts.’”

As its title suggests, Prakash’s essay briefly explains why math is vital to human nature. As the world continues to advance and modernize, society emphasizes sciences such as medicine, chemistry, and physics. All sciences employ math; it cannot be studied without math. It also helps us better our reasoning skills and maximizes the human mind. It is not only necessary but beneficial to our everyday lives.

## 4. Math Anxiety by Elias Wong

“Math anxiety affects different not only students but also people in different ways. It’s important to be familiar with the thoughts you have about yourself and the situation when you encounter math. If you are aware of unrealistic or irrational thoughts you can work to replace those thoughts with more positive and realistic ones.”

Wong writes about the phenomenon known as “math anxiety.” This term is used to describe many people’s hatred or fear of math- they feel that they are incapable of doing it. This anxiety is caused mainly by students’ negative experiences in math class, which makes them believe they cannot do well. Wong explains that some people have brains geared towards math and others do not, but this should not stop people from trying to overcome their math anxiety. Through review and practice of basic mathematical skills, students can overcome them and even excel at math.

“We see that math is not an obscure subject reserved for some pretentious intellectual nobility. Though we may not be aware of it, mathematics is embedded into many different aspects of our lives and our world — and by understanding it deeply, we may just gain a greater understanding of ourselves.”

Similar to some of the previous essays, Ali’s essay explains the importance of math. Interestingly, he tells a story of the life of a person name Kyle. He goes through the typical stages of life and enjoys typical human hobbies, including Rubik’s cube solving. Throughout this “Kyle’s” entire life, he performed the role of a mathematician in various ways. Ali explains that math is much more prevalent in our lives than we think, and by understanding it, we can better understand ourselves.

## Writing Prompts on Essays about Math

Math is a controversial subject that many people either passionately adore or despise. In this essay, reflect on your feelings towards math, and state your position on the topic. Then, give insights and reasons as to why you feel this way. Perhaps this subject comes easily to you, or perhaps it’s a subject that you find pretty challenging. For an insightful and compelling essay, you can include personal anecdotes to relate to your argument.

It is well-known that many people despise math. In this essay, discuss why so many people do not enjoy maths and struggle with this subject in school. For a compelling essay, gather interview data and statistics to support your arguments. You could include different sections correlating to why people do not enjoy this subject.

In this essay, begin by reading articles and essays about the importance of studying math. Then, write about the different ways that having proficient math skills can help you later in life. Next, use real-life examples of where maths is necessary, such as banking, shopping, planning holidays, and more! For an engaging essay, use some anecdotes from your experiences of using math in your daily life.

Many people have said that math is essential for the future and that you shouldn’t take a math class for granted. However, many also say that only a basic understanding of math is essential; the rest depends on one’s career. Is it essential to learn calculus and trigonometry? Choose your position and back up your claim with evidence.

Prasanna’s essay lists down just a few applications math has in our daily lives. For this essay, you can choose any activity, whether running, painting, or playing video games, and explain how math is used there. Then, write about mathematical concepts related to your chosen activity and explain how they are used. Finally, be sure to link it back to the importance of math, as this is essentially the topic around which your essay is based.

If you are interested in learning more, check out our essay writing tips !

For help with your essays, check out our round-up of the best essay checkers

Martin is an avid writer specializing in editing and proofreading. He also enjoys literary analysis and writing about food and travel.

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## Article Contents

1. introduction, 2. realism and explanatory indispensability, 3. dissecting ‘explanatory role’, 4. useful distinctions, 5. programme explanation, 6. the counterfactual account of explanation, 7. the kairetic account of explanation, 8. broader reflections.

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## On the ‘Indispensable Explanatory Role’ of Mathematics

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Juha Saatsi, On the ‘Indispensable Explanatory Role’ of Mathematics, Mind , Volume 125, Issue 500, October 2016, Pages 1045–1070, https://doi.org/10.1093/mind/fzv175

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The literature on the indispensability argument for mathematical realism often refers to the ‘indispensable explanatory role’ of mathematics. I argue that we should examine the notion of explanatory indispensability from the point of view of specific conceptions of scientific explanation. The reason is that explanatory indispensability in and of itself turns out to be insufficient for justifying the ontological conclusions at stake. To show this I introduce a distinction between different kinds of explanatory roles—some ‘thick’ and ontologically committing, others ‘thin’ and ontologically peripheral—and examine this distinction in relation to some notable ‘ontic’ accounts of explanation. I also discuss the issue in the broader context of other ‘explanationist’ realist arguments.

Much of the recent literature on the indispensability argument for mathematical realism discusses the ‘indispensable explanatory role’ of mathematics. It is commonplace to articulate the Quine-Putnam defence of mathematical realism in explicitly explanatory terms, and for many the prospects of platonism hang on the indispensability or otherwise of mathematics to empirical explanations. This paper identifies and addresses a clear lacuna in this prominent line of research: examining explanatory indispensability from the point of view of specific conceptions of scientific explanation. I will incentivize the task of filling out this critical lacuna, and also take some initial steps towards it by introducing a pertinent distinction regarding the way in which mathematics contributes to explanation. Drawing this distinction is critical, especially for the ontological conclusion of the indispensability argument.

To set the context I will review the significance of the notion of ‘explanatory role’ for the mathematical indispensability argument, first in general terms (§2), and then with reference to a particular mathematical explanation (§3). After drawing a distinction between ontologically committing ‘thick’ and ontologically peripheral ‘thin’ explanatory roles in general terms (§4), I will further examine this distinction in relation to three notable ‘ontic’ accounts of explanation: Jackson and Pettit’s programme explanation (§5); Woodward’s counterfactual account (§6); and Strevens’s kairetic account (§7). The connection between explanation and ontology is prima facie most plausible in the context of such ontic accounts of explanation, as opposed to epistemic or modal accounts, as I will explain (§4). In each case we will find ample room for regarding mathematics as playing only a thin, ontologically peripheral role in mathematical explanations of empirical phenomena. In conclusion, I will discuss the take-home message in the broader context of other explanationist realist arguments (§8).

Many realist arguments support ontological commitment to X by pointing to X ’s explanatory indispensability. This is exemplified by Baker’s ‘Enhanced Indispensability Argument’ for mathematical realism:

We ought rationally to believe in the existence of any entity that plays an indispensable explanatory role in our best scientific theories.

Mathematical objects play an indispensable explanatory role in science.

Hence, we ought rationally to believe in the existence of mathematical objects.

( Baker 2009 , p. 613)

This is of course just the familiar Quine-Putnam defence of platonism cast explicitly in terms of the ‘indispensable explanatory role’ of mathematics. 1 The key idea is that the indispensability argument should really turn on the specific kind of indispensability that commits a scientific realist to paradigmatic (non-mathematical) theoretical posits, such as electrons, quarks and black holes; arguably, the scientific realist is committed to such unobservables precisely because these ‘theoretical posits’ play an indispensable explanatory role in our best science.

If a belief plays an ineliminable role in explanations of our observations, then other things being equal we should believe it, regardless of whether that belief is itself observational, and regardless of whether the entities it is about are observable. ( Field 1989 , p. 15) 2

[A] great deal hangs on the role of mathematics in scientific explanations. If mathematical entities do not play the right sort of role in scientific explanations, then this needs to be spelled out in a way that distinguishes mathematical entities from other entities quantified over in our best scientific theories. ( Colyvan 2012 , p. 1043)

How do we best push the debate forward? Common sense suggests that we should aim to bring our best understanding of scientific explanation to bear on the debate, and examine the notion of ‘(mathematics’) explanatory role’ in relation to different analyses and conceptions of explanation. In this way we can hope to gain a much closer grip on ‘explanatory role’ so as to be better placed to judge whether mathematics plays the kind of explanatory role that actually matters for the ontological debate in question.

In the rest of this paper I will take some initial steps in this direction, by first looking into the notion of ‘explanatory role’ in general terms, before examining it in more detail in relation to three specific accounts of explanation. The premiss (1) of the Enhanced Indispensability Argument above is undermined as a result.

Let us briefly recall a central example of allegedly explanatory mathematics to begin with. (The rest of our discussion will benefit from having this staple example clearly in mind. Although I regard the example as problematic on multiple counts (see Saatsi 2011 ), for all its shortcomings and simplicity, it functions well to illustrate the points I wish to make, and it will also conveniently allow us to relate the present point of view to earlier literature.) 5

Explanandum : Why is the life cycle of the North American cicada thirteen or seventeen years (depending on subspecies)?

Stylized explanation of the seventeen-year period:

(4) Having a life-cycle period which minimizes intersection with other (nearby / lower) periods is evolutionarily advantageous. ( biological law )

(5) Prime periods minimize intersection (compared to non-prime periods). ( number-theoretic theorem )

(6) Hence organisms with periodic life cycles are likely to evolve periods that are prime. ( ‘mixed’ biological/mathematical law )

(7) Cicadas in ecosystem-type E are limited by biological constraints to periods from 14 to 18 years. ( ecological constraint )

(8) Hence cicadas in ecosystem-type E are likely to evolve 17-year periods.

( Baker 2009 , p. 614)

We can similarly explain the thirteen-year period by plugging in appropriate ecological constraints. The key biological premiss (4) can be supported in evolutionary terms, based, for instance, on the relative fitness advantage of individuals whose offspring are more likely to avoid predators that are themselves periodical and less likely to breed with other cicada species.

In this DN-type explanatory argument, mathematics can seem to play an explanatory role by virtue of being employed in the deduction of the explanandum (8). For someone operating with a Hempelian DN model in mind, the mathematical parts (5) and (6) above seem just as explanatory as the other parts, especially given the sense in which the universal ‘law’ (6) ‘covers’ the specific, ecologically constrained seventeen-year case stated in the explanandum.

The DN-model has long been obsolete, however, and it is furthermore unclear how ‘explanatory role’ should be linked to ontological commitment in something like the DN account. 6 But once we renounce the DN account, it is an open question how exactly explanatory contributions should be attributed to different parts of an explanatory story or argument (such as Baker’s stylized explanation above). It is thus very much legitimate for us to ask: are the mathematics-laden elements (5) and (6) really contributing to the explanatoriness of the explanation of (8) above?

How do we begin to answer this question? Naturally, by considering it in relation to different accounts of scientific explanation. (How else?) Before we get started on this, let us motivate this approach further by recalling some earlier commentaries on the cicada example.

(5/6)* For periods in the range 14 to 18 years the intersection minimizing period is 17. ( fact about time )

This line of thought can be accused of begging the question: what justification is there for recasting the mathematics involved as having a ‘merely’ representational role, if scientists themselves do not appeal to any kind of explanatory–representational dichotomy? Baker’s thesis that mathematics is indispensable for explaining can be directly motivated, by contrast, by taking scientists’ pronouncements to this effect at face value. This is in essence how Baker and Colyvan (2011) respond to the ‘indexing strategy’ (as executed in Daly and Langford 2009 ), according to which mathematics should only be viewed as ‘indexing’ (that is, representing) non-mathematical facts about the world. Baker and Colyvan maintain that the indexing strategy is at severe odds with the scientific practice which arguably indispensably involves number theory, for example, in connection with the periodical cicada. In a similar manner, Pincock (2012) defends the existence of mathematical explanations—explanations for which mathematics ‘makes an explanatory contribution’—via a ‘comparison test’, which considers available explanations of (say) the cicada phenomenon to see whether or not the ‘best explanation’ (as judged by scientists) makes use of a mathematical claim. 8

If we follow these authors in accepting the lead from scientific practice and admit that mathematics can indeed be indispensable to some of our best scientific explanations, is there any room left for me to manoeuvre? Is it not simply incoherent to now deny that mathematics plays an indispensable explanatory role?

Well, the answer to this question turns out on closer reflection to depend on what we mean by ‘explanatory role’. It now becomes critical for us to clarify this notion in order to zero in on the key point of contention. For while there undeniably is a sense in which we can construe mathematics in the cicada and other examples as ‘explanatory’, or as ‘playing an explanatory role’, what really matters for the indispensability argument—all that matters!—is whether or not mathematics plays the kind of explanatory role that we should take as ontologically committing . A properly informed analysis of scientific explanation is required in order to see whether a face-value reading of scientific practice (and the admission that mathematics is indispensable for explanations) implies that mathematics plays an explanatory role that is ontologically committing.

Instead of bluntly insisting that mathematics is not explanatory at all (or plays no explanatory role whatsoever), a critic of the explanatory indispensability argument is much better off insisting that mathematics does not play the right kind of explanatory role. This strategy requires, of course, that we can somehow distinguish between different types of explanatory roles, some ‘thick’ and ontologically committing, others ‘thin’ and ontologically peripheral. The next section will discuss this distinction in the abstract, along with some other useful distinctions. After that, I will show how this kind of distinction can be drawn in the context of some specific ontic accounts of explanation.

Salmon (1985) introduced the notion of an ontic account of explanation, distinguishing it from epistemic accounts, on the one hand, and modal accounts, on the other. This tripartite classification is well known in the philosophy of scientific explanation. It is worth recalling it here as its relevance has never been noted in the present context. For our discussion, a useful way of characterizing the classification is as follows.

The basic idea behind an ontic conception of explanation is that explanation is a matter of situating the explanandum within a broader ontic structure of the world. The explanatory power derives from stating some relevant worldly facts: objective causal or mechanistic facts, or nomological facts, or statistical relevance relations, or symmetries, or whatever ontic structures can bear an objective relationship of explanatory relevance to the explanandum. (Causal and causal-mechanical accounts of explanation are paradigmatic examples of the ontic conception, and also what Salmon primarily had in mind in characterizing this conception, but we should construe ‘ontic structure of the world’ in broader terms to make room for mathematical and non-causal explanations.) Typically, explanatory relevance is a matter of exhibiting some kind of dependence of the explanandum on the explanans, in the way ‘difference-making’ relations do in the paradigmatic case of causal explanation, for instance. But not all explanatory dependence is causal. A law can depend on other laws in an explanatory way, but laws do not cause other laws. An explanandum can depend on structural constitution in an explanatory way that is not causal, as in the case of glass’s fragility being explained by its molecular structure. An explanandum can depend on more abstract (yet still real) structural features of the world, as in the case of a Lorentz contraction being explained in relativity in terms of the fundamental kinematic structure of reality. We can also envisage the possibility of some kind of sui generis dependence of certain physical facts on some mathematical facts so as to make conceptual room within the ontic conception for ontologically committing mathematical explanations.

The basic idea behind an epistemic conception of explanation, by contrast, is that an explanation is whatever we give in order to explain, where explaining is an epistemic activity of providing understanding . Following Hempel and many others, providing understanding can be viewed as a matter of showing that the explanandum was to be expected (given the explanans), but different conceptions of understanding are possible. According to an epistemic conception, the explanatory power of an explanation derives from the fact that the explanatory information suitably entails the explanandum so as to provide understanding. It does not derive from the explanation pointing to features of reality that stand in a worldly relation of explanatory relevance to the explanandum.

Finally, according to Salmon, the modal conception of scientific explanation does its job by ‘showing that what did happen had to happen’ ( Salmon 1985 , p. 293). This is quite ambiguous by itself, and it is difficult to see any clear difference from the ontic conception: after all, situating the explanandum within a broader ontic structure of the world (as per the ontic conception) can show why the explanandum had to happen given that structure. Salmon thought the distinction between ontic and modal conceptions was a substantial one, but only in the context of indeterministic physics. Since this isn’t our present context, Salmon’s (somewhat inchoate) remarks on the modal conception (mostly made in the context of probabilistic and statistical explanations) are of little interest to us here.

What is of interest, however, is a recent characterization of the modal conception by Lange, according to whom ‘the modal conception, properly understood, applies at least to distinctively mathematical explanation in science, whereas the ontic conception does not’ ( Lange 2013 , p. 510). The modal conception, as Lange understands it, takes explanation to be a matter of showing that the explanandum is inevitable in the sense that it holds independently of any contingent ontic structure at stake.

[T]hat twenty-three cannot be divided evenly by three supplies information about the world’s network of causal relations: it entails [as a matter of mathematical necessity] that there are no [counter-legally possible] causal processes by which twenty-three things are distributed evenly (without being cut) into three groups. ( Lange 2013 , p. 496)

Let us now consider the issue of ontological commitment in relation to this rough-and-ready distinction between ontic, epistemic, and modal conceptions. Prima facie , the connection between explanatory mathematics and ontological commitment to mathematics is plausible in the context of an ontic account of explanation, since according to the ontic conception, explanatory power is derivative only from stating explanatorily relevant worldly facts. Given the characterization of ontic explanation, it is natural to think, prima facie, that if (purported) reference to X is indispensable for explaining, then X is real, as dispensing with X results in a loss of explanatory power, which is (purportedly) due to leaving out information about an ontic relationship of explanatory relevance between X and the explanandum.

If mathematical explanations are instead best construed in the fashion of either the epistemic or the modal conception, it is much less clear what the connection between explanation and ontological commitment is meant to be, and as a result explanatory indispensability arguments become immediately more contentious.

Consider the modal conception first. Here mathematics can play an explanatory role by virtue of being part of a derivation or deduction that shows that the explanandum was inevitable to a stronger degree than results from any assumptions regarding the ontic structure in place. Lange (2013) argues in a fascinating recent paper that this conception best captures (some) mathematical explanations in science. But Lange thinks, rightly in my view, that his modal account of mathematical explanations is largely orthogonal to issues of ontological commitment. In Lange’s analysis of ‘distinctively mathematical explanations’, mathematics plays an indispensable role in providing explanatory modal knowledge, namely, knowledge of the inevitability of the explanandum, in the sense of it not depending on the actual laws of nature. It is far from clear why this kind of explanatory role would give us grounds to construe mathematics realistically.

Assume for the sake of the argument that a given mathematical explanation—for example, the strawberry example above—explains by providing knowledge of the independence of an explanandum of the actual laws of nature, so as to show that ‘what did happen had to happen’ in a strong counter-legal sense of necessity. It is clearly a non sequitur to conclude from this that the mathematics affording us this knowledge must itself be given a realist interpretation. Admittedly, Lange writes in a way that lends itself to a realist reading, suggesting that mathematical facts are somehow responsible for the strong degree of necessity at stake. For example, ‘the mathematical fact entails that even a pseudoprocess rather than a causal process …cannot involve such a division of twenty-three things’ ( Lange 2013 , p. 496). But those who baulk at a realist interpretation of ‘mathematical necessity’ and ‘mathematical fact’ can happily accept Lange’s idea that mathematical explanations explain by showing that the explanandum is inevitable by virtue of being completely independent of some pertinent feature of the ontic structure of the world. According to Lange, the explanatory power resides in this modal information, so what really matters are the explanatory modal facts, and there is no clear prima facie connection between explanatorily indispensable mathematics and ontological commitment. The platonist may feel that an anti-realist will not be able to account for the indispensability of mathematics for obtaining such modal information. But it is not as if the platonist has at this point an explanatory upper hand by being able to say something informative about the connection between mathematical facts and modal facts. And I do not see any reason to think that mathematics as construed by a fictionalist, for example, would be unable to provide the kind of modal knowledge that matters according to Lange. 10

Now, let us move on to briefly consider the epistemic conception. Here an element of an explanation can be a source of explanatory power by virtue of its function in providing understanding. For example, if understanding is equated with nomic expectability, an advocate of the epistemic conception can take mathematics as explanatory if it is employed in an explanatory argument to show that (non-mathematical) laws and initial conditions entail the explanandum so as to demonstrate that it was to be expected. Playing an indispensable explanatory role of this sort could be a matter of calculational support, say; perhaps nomic expectability of the explanandum would otherwise be only apparent to a Laplacean demon, for instance. It is far from immediately clear why this conception of understanding—or other conceptions, for that matter—would give us grounds to construe mathematics realistically. 11

Thick explanatory role is played by a fact that bears an ontic relation of explanatory relevance to the explanandum in question. Thin explanatory role is played by something that allows us to grasp, or (re)present, whatever plays a ‘thick’ explanatory role.

This question, I submit, can be properly answered only in the context of specific accounts of explanation, which also allow us to make clearer and more precise the distinction between thick and thin explanatory roles. I will now turn to these accounts.

Lyon (2012) is one of the few to analyse the explanatory role of mathematics in terms of a specific account of explanation. He argues, in effect, that mathematics can be seen to play (what I have now called) a thick explanatory role: it plays a ‘programming’ role as characterized by Jackson and Pettit as part of their model of programme explanation. In examples like the one involving periodical cicadas, ‘mathematics is indispensable to the programming of the efficacious properties’ ( Lyon 2012 , p. 568).

Elsewhere, I have criticized Lyon’s attempt to cast mathematical explanations as programme explanations ( Saatsi 2012 ). In the context of our current discussion, the key point of this criticism can be expressed as follows: Lyon has not established that mathematics indeed plays a thick explanatory role in the context of the programme explanation account. That is, Lyon’s account leaves fully open the possibility that the mathematics involved in empirical explanations, while indispensable, could be construed as playing merely a thin explanatory role.

To see this problem clearly, let us briefly recall Lyon’s argument. Jackson and Pettit (1990) introduce the notion of programme explanation by contrasting it with a process explanation , which is a fine-grained, straightforwardly causal explanation. A programme explanation is more abstract, and is causal only in a derivative sense: it cites a property which guarantees the instantiation of some causally efficacious property involved in a process explanation of the same explanandum. According to Jackson and Pettit, a liquid’s temperature T , for example, can (programme-)explain the cracking of a glass receptacle by virtue of playing a programming role with respect to a lower-level property also instantiated by the liquid: some particular molecules hitting the receptacle wall with a high enough collective momentum to cause the cracking. 13 Jackson and Pettit use examples like this to depict a sense in which a higher-level property (such as T ), which is not causally efficacious itself, can nevertheless be explanatorily (and causally) relevant by virtue of standing in a suitable ‘programming’ relationship to some causally efficacious lower-level property.

The programme explanation account is an ontic one: a successful explanation of an event E requires that we point to a feature of the world that is explanatorily relevant by virtue of there being a sense in which E depends on that feature of the world. The properties that are explanatorily relevant in this sense, by virtue of ‘programming’, play a thick explanatory role characterized by an intimate modal relation between the higher-level programming facts and the lower-level causal facts. (Basically, it is not possible to have an explanatory higher-level fact without also having one or another causally efficacious lower-level fact. In many examples, this relation between the properties is some kind of ‘realization’ relation.) Jackson and Pettit capitalize on this modal relation to argue that a higher-level property provides information about what the explanandum depends on, going beyond the information provided by the corresponding lower-level process explanation. 14

They cite properties and/or entities which are not causally efficacious but nevertheless programme the instantiation of causally efficacious properties and/or entities that causally produce the explanandum. And importantly, they cite mathematical properties and/or entities that are doing (at least part of) this programming work. ( Lyon 2012 , p. 567)

[I]f we take away any mention of primeness from the cicada explanation, the explanation falls apart, and there doesn’t seem to be anything that would put it back together. ( Lyon 2012 , p. 568)

There are various difficulties in construing mathematics as playing a programming role (see Saatsi 2012 ). A key problem is the following. For Jackson and Pettit, ‘programming’ is a matter of a modal relation holding between higher-level (‘programming’) and lower-level (causally efficacious ‘process’) properties. 15 It is critical to have a grip on this modal relation in order to grasp the sui generis explanatory dependence at stake: programme explanations are explanatory by virtue of providing information about a higher-level dependence rooted in the modal relation between higher- and lower-level properties. But we have no such grip at all for the alleged modal relation that is meant to hold between a mathematical fact and the lower-level causal facts.

An alternative, and perhaps better, way to express the problem is this. An advocate of the enhanced indispensability argument needs to establish that mathematics is playing a thick , ontologically committing explanatory role; establishing indispensability simpliciter is not enough. In the context of the programme explanation idea, this can only be done by providing a substantial account of the modal relationship between the mathematical features that allegedly ‘programme’, on the one hand, and the properties that feature in the corresponding process explanations, on the other. Lyon does not provide anything to this effect, however; rather, he just stresses the explanatory indispensability of mathematics in explanations that he construes as programme explanations (see, for example, the quote above). But this is not enough, as indispensable mathematics here could instead be interpreted as playing a thin, ontologically peripheral role.

Lyon’s appeal to the de facto indispensability of mathematics is similar in spirit to Baker and Colyvan’s appeal to scientific practice. Keeping in mind the need for a thick explanatory role and the details of the programme explanation account, we can see why they all achieve little by such an appeal; the explanatory practice of science does not wear its ontological commitment on its sleeve.

If mathematics does not play a programming role, but is nevertheless explanatorily indispensable, what other roles are there? A natural alternative is to argue that mathematics only plays the role of representing some non-mathematical features of the world that themselves play the programming role (as I suggested in Saatsi 2012 ). The challenges of spelling out this suggestion depend on the case at hand, and one should not treat lightly the complexities involved in some important cases (see, for example, Batterman 2010 and Wilson 2013) . 16 But in the cicada case, at least, mathematics arguably can be construed as playing such a thin representational role, representing features of time (such as that stated in (5/6)*, above, and natural generalizations thereof) that play a thick explanatory role ( Saatsi 2011 ).

Jackson and Pettit’s programme explanation model is not a full-blown theory of explanation. Rather, it is an attempt to make sense of higher-level explanations that do not feature causally efficacious properties. Its core idea of explanatory modal dependence can arguably be incorporated into a broader account of explanation due to Woodward (2003b ), so it is worth examining the explanatory role of mathematics in the context of this account. 17

Woodward’s counterfactual theory of explanation is a prominent contemporary account in the ontic tradition. Woodward primarily intended his account to capture causal explanations, but various people have argued that its central idea can also be applied to non-causal (e.g. geometrical) explanations ( Bokulich 2008 , 2011 ; Saatsi and Pexton 2013 ; Ylikoski and Kuorikoski 2010 ). 18 Hence, there is no reason to dismiss it as irrelevant in relation to (prima facie) non-causal explanations, for example, of cicada periods.

[A]n explanation ought to be such that [it enables] us to see what sort of difference it would have made for the explanandum if the factors cited in the explanans had been different in various possible ways. ( Woodward 2003b , p. 11)

For Woodward, a DN-type deduction is only explanatory to the extent that it provides such modal information. The explanatoriness (or ‘explanatory power’) of an argument or derivation springs from correctly representing the relevant objective dependency relations in the world, grounded in the world’s actual (causal-)nomological structure. 20 The account thus accords with the ontic conception of explanation. The explanatory dependency relations can furthermore be found at different ‘levels’. For example, there is a dependency relation that connects the cracking of the glass receptacle to the liquid’s temperature, allowing Woodward to conceive of the temperature as a cause of the cracking (in Woodward’s precise sense of causation).

[My account] is less demanding about the need for explanatory theories to have a defensible ‘realistic’ interpretation (in the sense of postulating only ‘real’ entities) and assigns a more prominent role to ‘instrumental’ success than many competing theories of explanation. …[S]uch an account fits explanatory practice in many areas of science much better than more ontologically oriented alternatives. ( Woodward 2003b , p. 232)

In Woodward’s account there is scope for regarding mathematics also as playing a thin explanatory role, on a par with Newtonian gravity. Whether or not the mathematics employed in scientific explanations is true in a realist sense, mathematics can be explanatory by virtue of allowing us to grasp physical dependency relations that play a thick explanatory role and are ultimately fully responsible for the explanatoriness of an explanation.

Consider the cicadas again. Explaining this phenomenon in a Woodwardian framework is a matter of grasping how the fitness-maximizing life cycle of a given cicada species—construed as a variable that can take different values—depends on other biologically relevant variables, such as other cicada species’ and predator species’ life cycles. Even if mathematics is indispensably involved in grasping or representing these explanatory modal facts, in Woodward’s account it is only the modal facts themselves that are ultimately the source of explanatory power. An advocate of this account could thus agree with Baker, Colyvan and Lyon that number theory is indispensable for a maximally ‘robust’ or ‘general’ explanation of the cicada periods, without viewing the numbers themselves as playing a thick explanatory role. Rather, number theory would be viewed as an indispensable vehicle for grasping the relevant modal facts: roughly speaking, in the long run the evolution of a prime-numbered period of p years depends only on there being competing/predator species of nearby periods and ‘ecological constraints’ that appropriately limit the range of viable possibilities (cf. §3). 22

One may question the depth of the analogy between Newtonian gravity and mathematical explanations: Newtonian gravity is not, of course, really indispensable for explaining gravitational phenomena, given that we can (a) formulate Newtonian gravity geometrically (as the Newton-Cartan theory), so that the explanatory modal relations are captured by the curvature of four-dimensional spacetime instead of gravitational force, and (b) supplant the theory by the general theory of relativity, which is explanatorily deeper by virtue of encompassing a larger set of explanatory dependence relations. 23

In response, it can be argued that the indispensability or otherwise of a theoretical posit is neither here nor there regarding the key issue at stake: in Woodward’s account, a theory can be truly explanatory even if its ‘fundamental ontology’ is only playing a thin explanatory role. A right-minded advocate of the account would not infer the reality of gravitational force from its explanatory indispensability, even if Newton’s theory had not been supplanted by general relativity, and even if the practising scientists preferred gravitational force explanations over a cumbersome Newton-Cartan formulation.

What must be admitted, however, is that this instrumentalist aspect of Woodward’s account is a double-edged sword for a scientific realist who wishes to link explanatory indispensability to ontological commitment for quarks, electrons, and other non-mathematical theoretical posits: these can, of course, be similarly construed as playing only a thin explanatory role. Therefore any ‘explanationist’ argument for realism about quarks, electrons, and so on, must incorporate more than mere appeal to explanatory indispensability. (I will reflect on this issue further in the concluding §8.)

Woodward’s account of explanation is not the only game in town: there are, of course, competing theories in the ontic spirit. While not aiming to offer a comprehensive review, I will briefly consider one prominent alternative next: Strevens’s (2008) kairetic account.

Strevens’s (2008) theory of explanation provides a criterion of explanatory relevance as a matter of difference-making : to explain an explanandum is (roughly speaking) to say what made a difference to its taking place or having the features it has. I am not going to delve deeply into the rich details of Strevens’s ‘kairetic’ account of difference-making. My objective is rather to make the point that in this ontic account too a distinction between thick and thin explanatory roles can be drawn so as to enable us to divorce explanatory indispensability from ontological commitments.

At the heart of Strevens’s account is the idea that explanatory difference-makers can be specified by abstraction from the specific details of fundamental causal relations (as well as laws and background conditions). 24 The abstracted difference-making facts are facts about nomological dependence (that Strevens understands as causation); thus, Strevens’s is an ontic account. As with Woodward, a DN-type explanatory deduction is only explanatory to the extent that it provides information about such worldly dependence relations. (Strevens is metaphysically more committed than Woodward: an explanatory deduction must ‘mirror’ worldly relations of explanatory relevance grounded in fundamental causal relations.)

Specifying difference-makers through abstraction has some remarkable pay-offs. For example, Strevens uses it to give an account of the explanatory role of deliberate falsehoods involved in idealizations (such as the ideal gas model explanation of Boyles’s law). Of particular interest to us is the way in which abstraction can also naturally lead to mathematical explanations in science, explanations in which mathematics is more than a derivational handmaiden, by virtue of conveying ‘understanding that turns on the appreciation of a central fact that is a matter of mathematics alone’ ( Strevens 2008 , p. 301). That is, through abstraction mathematics can acquire a bona fide explanatory role. But at the same time Strevens explicitly states that ‘[one does not need] metaphysical assumptions to spell out its additional contribution to explanatory goodness’ ( Strevens 2008 , p. 304). In other words, Strevens denies that mathematics, even when indispensable, gets to play a thick, ontologically committing explanatory role.

First, the mathematical structure of an explanatory derivation must reflect the corresponding relations of causal production in the world. Only by grasping the mathematical structure do you follow the process of production from beginning to end. Second, the mathematics of the derivation tells you implicitly, and sometimes explicitly, what makes a difference to the causal production and what does not, and why it does or does not. ( Strevens 2008 , p. 329)

The second, more interesting way for mathematics to contribute is exemplified by the cicada example. A detailed reconstruction and analysis of the cicada case in the context of Strevens’s account is beyond the confines of this paper, but we can capture the gist of the matter by saying that in a Strevensian framework the mathematics functions to show how the explanandum is independent of the low-level causal details. 25 Thus in the stylized cicada explanation (§3) the derivation of (6) employes number theory so as to allow us to grasp the fact that almost nothing about the causal trajectories of individual cicadas makes a relevant difference to the long-run tendency to evolve a period that is co-prime to every ecologically available alternative.

No philosophy of mathematics, note, is invoked. I assume only that mathematics is somehow able to serve as a representer of things in the world; this is consistent both with the view that the world is inherently mathematical and with the contrary view. ( Strevens 2008 , p. 330)

The principal aim of my discussion has been to make a simple but crucial methodological point: debates about the ‘explanatory role’ of mathematics (in the context of the indispensability argument) should be conducted much more closely in relation to specific accounts and conceptions of explanation. There are fruitful, more fine-grained discussions to be had about the various interesting cases of explanatory mathematics in the context of the recent ontic accounts of explanation developed by Strevens and Woodward, for example, and in the context of the modal account developed by Lange. In relation to these specific accounts, I have argued that the advocates of the enhanced indispensability argument have fallen much short of having shown that mathematics, even when indispensable, plays the kind of thick explanatory role that is ontologically committing. (I believe there are further distinctions between ontologically committing and ontologically peripheral explanatory roles to be drawn in connection with various epistemic and modal accounts of explanation as well, and there is therefore much further work to be done to defend the indispensability argument, whichever account of explanation is preferred.)

I would now like to suggest more generally that in other philosophical contexts too, arguments from explanatory indispensability would considerably benefit from precisifying the intuitive notion of explanatory indispensability in terms of a philosophical conception of explanation. My argument has taken place purely in the context of the indispensability argument for mathematical realism, but there is an obvious broader context for it. Consider the debate about scientific realism to begin with. The (explanatory) indispensability argument has been designed to mimic similar arguments for scientific realism, and there is indeed a long tradition in philosophy of science, going back to the very architects of the indispensability argument, to argue for scientific realism by appeal to the explanatory role that theoretical posits like quarks and electrons play in science. If an argument for scientific realism about quarks or electrons is nothing but an explanatory indispensability argument applied to these theoretical posits, it should clearly be supplemented with a justification for thinking that the relevant theoretical posits are playing a thick explanatory role. It is an interesting question exactly when this supplement can be provided, and what role (if any) causation should play. 26 Not all realist arguments can be readily assimilated to the explanatory indispensability argument, however, and even in the case of the realist’s most famous inference to the best explanation—the no-miracles argument—it has been argued that it can be naturally furnished with fine structure that critically differentiates it from the abstract explanatory indispensability argument (see Saatsi 2007 ).

The only workable criterion of reality is the explanatory criterion: something is real if its positing plays an indispensable role in the explanation of well-founded phenomena. ( Psillos 2011 , p. 15)

It is easy to find other explanation-driven arguments in other areas of metaphysics that bear a close resemblance to these examples from philosophy of science. Some Australasian metaphysicians, like Armstrong and Sankey, frame their arguments for universals and natural kinds in terms of (something like) explanatory indispensability. As far as I am concerned, such arguments have little probative force without a proper analysis of the sense of explanation in play. Going even further afield, we find instances of (something very much like) the explanatory indispensability argument in areas such as metaethics, where we have perhaps even less of an idea what explanation amounts to exactly. 28

There is a general, commonsensical point to be made about all such arguments: ultimately a handle is needed on the concept of explanation involved, for how else can we really assess the epistemic and ontological import of explanatory indispensability, if not by first spelling out what it takes to explain something?

1 The indispensability argument in the Quine-Putnam tradition turns on a broader notion of indispensability, typically associated more closely with predictions and confirmation. In the more recent literature there has been a distinct shift towards the explanationist construal of indispensability.

2 It is clear from the quote’s context that ‘unobservable entities’ for Field can include mathematical abstracta, and an important part of Field’s (1989) defence of fictionalism is his denial that mathematics ever plays such an ‘ineliminable’ explanatory role.

3 This both begs the question and is in tension with the actual science, which is seemingly teeming with non-causal explanations.

4 Baker (2005) contains a brief discussion of three broad accounts of explanation. Lyon (2012) is a much more notable exception, which I will presently examine (§5). See also Batterman (2010) and Pincock (2011) for useful analysis of the explanatory role of mathematics in the context of the so-called mapping account of applied mathematics. Apart from these few exceptions, the growing literature on the indispensability argument just floats free of theories of explanation. For example, while Baker (2009) ably analyses (and defends) the notion that mathematics is indispensable for some of our best scientific explanations, he says little in general about what it is to play an explanatory role, and none of his analysis relates to a considered theory of scientific explanation. Similarly—on the other side of the debate— Yablo (2012) has recently argued that the indispensability of mathematics in scientific explanations can be interpreted in ways that do not support the indispensability arguments, since the explanatory role need not be played by mathematical objects . Yablo’s point is, in my view, well taken, but again it is worth pointing out that his analysis of the explanatory role of mathematics is given in general terms, and not in relation to any specific account of scientific explanation.

5 Lange (2013) refines the conception of the cicada explanation as a mathematical explanation. I will stick to Baker’s original presentation to better reflect the evolving dialectic around it.

6 Although it is part of the ‘adequacy conditions’ in Hempel’s DN model that for any actual explanation the explanans must be true , the problem is that the DN account is an ‘epistemic’ (as opposed to ‘ontic’ or ‘modal’) account of explanation. (I will recall the distinction presently.) According to the DN model the function of an explanation is to give understanding . (Hempel equated understanding with expectability , but other conceptions of understanding could be conjoined with the central idea that explanations are arguments that provide understanding.) However understanding is construed, it is not clear why mathematics, fictional models, or even things like diagrams would have to be true in order to play an explanatory role in the epistemic sense of providing understanding.

7 More specifically, I argued that the inclusion of mathematics in the cicada case is not a source of any further explanatory power, and that the prime number explanation does not enjoy any advantage of generality, since (5/6)* can be naturally extended to more general facts about time.

8 Note that Pincock does not endorse the inference from explanatory indispensability to mathematical realism, however, since he denies the principle of inference to the best explanation in the form required for this inference.

9 Note that an explanation conforms to the modal conception only if it provides such explanatory modal information without simultaneously providing ontic information about some kind of dependence relation connecting the explanandum to the relevant mathematical facts or properties. If an explanation provides such modal information by virtue of showing, for example, how the explanandum only depends on the system instantiating certain abstract mathematical properties, then it should be viewed as ontic.

10 Note also that mathematics can be used to obtain explanatory modal information even when it is not indispensable. The strawberry example is a case in point, since it can be paraphrased into an argument in first-order logic entailing the explanatory modal fact without referring to numbers.

11 There is an interesting connection here to issues concerning the question of whether understanding is factive: several philosophers have argued that falsehoods (such as idealized models) can function to provide understanding, and even be indispensable for it. See, for example, Elgin (2004) .

12 I do not pretend to have shown that the debate is a non-starter in the context of modal or epistemic conceptions. But while further work remains to be done in this context, we can appreciate the corresponding lacuna in the indispensability debate just on the basis of our broad characterization of these two conceptions.

13 See Jackson and Pettit 1990 for details.

14 For example, arguably the dependence of the cracking of the container on the liquid’s temperature goes over and above the information about the way in which the cracking depends on some particular set of molecules having specific momenta.

15 For the paradigmatic examples, the programming relation is one of ‘realization’ (broadly construed): temperature, as mean kinetic energy, is realized in different molecular configurations. A determinable property is realized in different determinates, and so on.

16 But note that neither Batterman nor Wilson explicate the non-representational explanatory role of mathematics in terms of any account or conception of explanation. It is an open question what the details of these cases will look like in the context of an appropriate account of explanation.

17 Ylikoski (2001) argues in detail that Woodward’s account (or something very much like it) can do all the programme explanation model can. See also Woodward (2009).

18 Woodward’s sympathies to this possibility are evident in Woodward (2003b , §5.9).

19 See references above. Woodward (2003b , §5.9) gives the example of explaining the stability of planetary orbits in terms of its counterfactual dependence on the dimensionality of space.

20 As Woodward puts it: ‘It is physical dependency relations …that are primary or fundamental in causal explanation; derivational relations do not have a role to play in explanation that is independent or prior to such dependency relations, but rather matter only insofar as (or to the extent that) they correctly represent such relationships’ ( Woodward 2003b , p. 211).

21 See Woodward (2003a) . Harper (2011) explores in great detail the surprising richness of the modal relations encapsulated in Newton’s theory.

22 An explanation that captures this ‘robustness’ or ‘generality’ of the prime numbered periods with respect to historico-ecological details can be viewed in Woodward’s account as maximizing a particular dimension of ‘explanatory depth’, measured by the number of what-if-things-had-been-different questions. (See Hitchcock and Woodward 2003 .)

23 This theory captures a larger set of explanatory dependencies, since it is applicable to a wider range of situations, for example, those involving very large masses and velocities.

24 The intuitive idea can be illustrated as follows. A hooligan throws a rock at a window. What made a difference to the complete shattering of the window is a complex combination of the projectile’s hardness, momentum and manner of impact together being ‘enough’. More specific facts regarding the missile’s velocity or mass, say, should not be construed as difference-makers in and of themselves, since we could vary any of those precise facts without altering the outcome (as specified by the explanandum). There is an unduly complicated story of fundamental physics about the precise causal mechanism involved. The explanation of breaking does not explicitly involve the details of this story; rather, the difference-maker is an abstract amalgamation of the object’s hardness, momentum and manner of impact.

25 Strevens does not consider the cicada example, but for comparison one can look at his evolutionary example of homozygous elephant seals, which has some of the same features.

26 It is not difficult to conceive of potentially relevant differences between mathematical and non-mathematical theoretical posits. There are many thinkable differences in the explanatory roles played by imaginary or prime numbers and quarks, for example, that do not boil down to the causal–non-causal contrast. The simple idea that ordinary observable matter is in some sense composed of quarks, for example, could perhaps serve as a starting point.

27 For example, Psillos argues that a scientific realist should countenance the reality of (non-mathematical) abstract objects like a system’s centre of mass because it can play an indispensable explanatory role (despite being causally inert). Now, if one examines such an explanation in Strevens’s account, for instance, the centre of mass naturally comes out as an explanatory difference-maker. But given the details of the account, it does so in a way that need not elicit a realist reification of it over and above of the more fundamental facts by virtue of which it is a difference-maker.

28 See Sturgeon (2006) and Enoch (2013) for discussions of explanation as a guide to the epistemology and metaphysics of moral realism.

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