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I believe in the beginning of Human race, mathematics emerged as a necessity so simplify descriptions of the things we observe in our daily life. But as of the last few centuries, it seems so Mathematics has departed from this physical utility perspective and moved to a sort of "a thing with it's internal logic which can be studied by external observer", so in this case, what reasons could be given on why mathematics should be studied?

It seems that utility is no longer the focus and I doubt many mathematicians would agree that the reason they study math is due to "aesthetics" as when one decides to be precise, there is not much more room for creative license.

I address a few points repeatedly bought up by users to my question.

  1. Many mathematics field has application today eg:Number theory in cryptography

I believe the number theory came first before the cryptography. The cryptography applications of NT was more like an afterthought than the original goal. I think it is clear that mathematics found in the pure field in modern day has applications only much later.

Furthermore physical applications of mathematics is actually nothing about mathematics because it often a very dirty step to phrase physical reality in terms of something which mathematics can be applied. One would often require a lot of approximations and simplifying assumptions to make it possible.

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    "in the beginning of Human race, mathematics emerged as a necessity so simplify descriptions of the things we observe in our daily life" Not necessarily: since the beginning in Ancient Greece, the study of axiomatic geometry as well as the theory of numbers are devoid of "daily life applications". Sep 1, 2022 at 13:27
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    And also your view about "modern mathematics" is wrong: consider e.g. the "useless" number theory, that is nowadays used for cryptography. Sep 1, 2022 at 13:28
  • I mean how does cryptography being applied applied NT contradict what I said? I just said application is secondary.I was of impression arthimetic was the first historically. Sep 1, 2022 at 13:28
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    As a published, currently active mathematician, I do mathematics because it allows me to be creative and, above all, because I find it stunningly beautiful. Sep 1, 2022 at 19:03
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    Your impressions are incorrect pretty much across the board. The explosion in recent centuries is largely driven by applied needs, from calculus to service mechanics, to functional analysis to service quantum mechanics, and to geometric analysis to service gauge theories. And, believe it or not, even many applied mathematicians talk about aesthetics, elegance and creativity as a major motivation. That precision leaves little room for creativity is false even outside of mathematics, much of high art, be it painting or music, owes its intricacy and sophistication to precision.
    – Conifold
    Sep 1, 2022 at 19:55

7 Answers 7

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The same analysis will apply to many forms of academic endeavor, such as theoretical science, philosophy, arts, ancient history, and some others I have neglected due to my faulty memory.

There are many types of motivation for studying such topics. And many different motivations for support from the wider community. No single motivation is going to explain the entire activity.

Aesthetics will certainly motivate some people. There are also related and overlapping enjoyment features. Some people love puzzles, for example. Some people love to seek knowledge, especially new knowledge. Some people love to create their art. Generically, such enjoyment arises because humans have "big brains" and it is, for many, enjoyable using them. Different abstract studies will have different components of these related attractions.

And the enjoyment will result in many people wanting to give support to such study, even if they cannot do it professionally. This is the reason the Perimeter Institute exists. A guy became an engineer, invented a type of cell phone, and made a ton of money. But his first academic love was always theoretical physics. So he endowed them with a bunch of money. Then he got his business partners to add to the endowment. Then he persuaded the province to add more. And now, the place is pretty secure.

Practical application is a motivating factor even for the most seeming abstract academic of studies. As an example that is comforting to me (due to my background in theoretical physics) consider Grassmann numbers. This is a very unusual algebra of numbers that have the peculiar property that their square is zero. It turns out these are useful in understanding the behavior of elctrons. And in turn, that is useful in understanding certain physics experiments. Which in turn are useful in understanding certain types of electronics. Which in turn helped advance the study of semiconductors. Which you are presumably looking at to read this on some type of computer.

Or, to take a quote form Isaac Asimov:

There is a single light of science, and to brighten it anywhere is to brighten it everywhere.

Another motivation is as a way to detect and nurture ability. One place wants cryptographers. Another wants computer programmers. Another wants people with the ability to understand enormous databases of seemingly unrelated data. Another wants to detect and predict trends. Another wants to understand the behavior of groups under various stress conditions. The ability to do abstract maths is useful to predict ability at these other tasks. The mental processes to do maths will train one to understand and solve a wide variety of problems. And those problems have application in a wide variety of applications.

Such motivations also have some ability to get support for study other than math. History, for example, can often help a military officer understand an opponent. Art theory can contribute to understanding marketing. Linguists can contribute to making computer interfaces easier to use. And so on over nearly any academic study.

There are, of course, less rarified motivations. Abstract academic studies have prestige and garner respect. Various granting agencies and political groups are often motivated by this prestige. So a politician may find it useful to advocate for grants to be given to a university including the departments doing such abstract studies. Such support does not necessarily flow in any strong relationship to anybody's evaluation of the worth of such studies. It flows in relation to a politician's perception of the prestige it produces for him.

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  • Well written. Though as a Waterloo resident, I worry that PI hasn't been able to attract good talent, but rather has had to pay decent-but-far-from-unique people absolutely insane salaries in order to get them to risk working for at such an institute (and that most of the people listed as being affiliated with PI are not actually there, yet are getting paid insane amounts and given abundant resources or office space). Sep 3, 2022 at 23:45
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I don't know what the motivation is, but:

We needed non-euclidean geometry to get general relativity, and we need general relativity for GPS. When people were thinking about non-euclidean geometry, they weren't thinking about relativity OR GPS, but here we are.

Understanding and discovering things is intrinsically rewarding to the individual. Society pays for people to understand and discover because because it's an investment in our future.

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  • We don't need relativity for GPS. Sep 2, 2022 at 21:20
  • @user1271772 Yes, we do need general relativistic corrections for GPS. Sep 2, 2022 at 22:29
  • Not according to this. Sep 2, 2022 at 22:37
  • @user1271772 True. A person on SE makes this claim in response to an article from a physics department. But according to the article cited by that user, and according to NASA (nasa.gov/mission_pages/chandra/images/…), we do. A quick search reveals that "alternative physics" sites and random blog articles are where we find the claim that GR doesn't affect GPS, and physics sites consistently state that it does. The best evidence I am able to find supports the assertion that GR is used to correct GPS systems.
    – philosodad
    Sep 3, 2022 at 2:21
  • @philosodad "because NASA said so" does not mean much. No one here said "GR doesn't affect GPS". If you implement GPS in a naive way (e.g. with only 1 satellite) you would need to incorporate GR into the calculations in order to avoid errors, which are actually noticeable in real-world applications. However the way that GPS is actually implemented is more sophisticated, involving multiple synchronized satellites at the same altitude. It might be fashionable to say GR is "needed" for GPS, and I wish it was, because that would be cool, but the reality is that GPS can work without it. Sep 3, 2022 at 5:03
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The most basic mathematical skills like:

  • the recognition of parts, numbers and their properties like commutativity and associativity

  • as well as the geometry of space, for example, recognising straight lines

are part of our deep visual grammar, pretty much like the deep grammar that Chomsky says underlies human language and which is determined by our biology. This does not mean that we need be conscious of this "deep mathematical grammar" and most people in this world aren't in the same way that most people know how to speak and learnt how to speak without ever picking up a grammar book.

Personally, I think that conscious mathematics didn't arise out of necessity but out of ritual, magic, play and religion. Conscious mathematics is after all symbolic. And the very first stirrings that we see of recorded symbolic language - prehistoric cave art - has nothing to do with necessity even if they described things of necessity like hunting.

Having said all this, physicists since the mid-20C have said that mathematicians have departed into their own wonderland of labyrinthine abstractions that few can follow, understand, or motivate. And as physics of all the sciences is closest to mathematics, what hope is there for the rest of us? Not only that, it's notorious that even within mathematics, even other mathematicians find each others' papers difficult to understand. Still, I would ascribe this to the very success of mathematics such that there has been an exponential growth in its methods and means, so much so that even professional mathematicians find it hard to keep up.

But there is hope too. Physicists in the 70s realised that fibre bundles that mathematicians had developed were just the right tool to geometrise gauge theory. And that homotopy theory, complex geometry, and differential geometry were important tools to make geometric sense of constructions in string theory and quantum field theory.

Still, this is far removed from the experience and interest of most people, and somehow I don't think that this is going to change very much in the near or far future despite all the pretty fractal pictures and photos of starburst galaxies that the James Webb telescope has shown us and developed, in part, by the very mathematics you are disparaging.

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The question has a number of wrong assumptions. A large amount of modern math does have physical applications. For example, the recent Nobel Prize based on topological insulators. There is the entire branch of cryptography, underpinning the entire internet. Even metaphysical math philosophy can trickle back to the "real" world; you can make different philosophical arguments about large cardinals, and this has repercussions back to certain kinds of computer programs (see: The relationship between large cardinals and program termination ).


As for aesthetics, "beauty is in the eye of the beholder." But there are certain symmetries, and asymmetries, and patterns that are buried in deep abstractions. Erdos referred to this source of deep math beauty as The Book. Tell me one of the biggest proofs of the last few years was not a thing of beauty:

I was sitting at my desk examining the Kolyvagin–Flach method. It wasn't that I believed I could make it work, but I thought that at least I could explain why it didn't work. Suddenly I had this incredible revelation. I realised that, the Kolyvagin–Flach method wasn't working, but it was all I needed to make my original Iwasawa theory work from three years earlier. So out of the ashes of Kolyvagin–Flach seemed to rise the true answer to the problem. It was so indescribably beautiful; it was so simple and so elegant. I couldn't understand how I'd missed it and I just stared at it in disbelief for twenty minutes. Then during the day I walked around the department, and I'd keep coming back to my desk looking to see if it was still there. It was still there. I couldn't contain myself, I was so excited. It was the most important moment of my working life. Nothing I ever do again will mean as much.

Andrew Wiles, quoted by Simon Singh via https://en.wikipedia.org/wiki/Wiles%27s_proof_of_Fermat%27s_Last_Theorem

If you find pictures more aesthetically pleasing than ideas, then there are pages such as https://mathoverflow.net/questions/178139/examples-of-unexpected-mathematical-images and links contained such as https://math.ucr.edu/home/baez/roots/ .


The last incorrect assumption is that there is no creativity. Except for very specific cases in Logic, there is almost always more than one way to prove a result. Often, notable proofs are refined later as either a more concise proof, or with fewer restrictions, or with smaller necessities. All of these variations require creativity to obtain.

And besides that point, the entire point of education for post-graduate level university (and sometimes even before) is to publish novel results. (And also pursued by people outside of academia). But if the goal is novel results, then you must solve a problem that no one has solved before. And this requires creativity.

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  • I mean my point was that the physical concept of topological insulators didn't lead to topology. The pattern nowadays is sott of like some math comes, much time passes and application js found in comparison to before where math had always some direct connection to the reL world. Sep 2, 2022 at 12:01
  • Topological insulators is not topology. This was new recent work to explain a physical phenomena. The same for Maxwell's work at the time. Public key cryptography was the instantiation of an application. These all have direct connections to the "real" world. Most math even has indirect connections to the real world, like I mention above. If you are trying to ask a question like "Why does anyone study anything other than Newtonian physics" then you need to clarify.
    – BurnsBA
    Sep 2, 2022 at 12:45
  • Well I mean whatever topological ideas they were based on. On the contrary to mathematics, physics has a clear progress criteria/ motivation , at least to me. It is gain the ability to predict the future, as the basis for the theories if it agree with predictions or not.@BurnsBA Sep 2, 2022 at 12:47
  • It seems you have two related questions, 1) about broad paradigm shifts and 2) about teleology and/or philosophy of science. Neither of these aspects are present in your initial question. If that's what you're interested in, I would open a new question asking about what you say in this comment, because it doesn't sound like you're really asking why people study math, you have a deeper kind of question.
    – BurnsBA
    Sep 2, 2022 at 12:58
  • I believe they are indirectly inside, because the broad paradigm shifts relates to utility thing vs "thing studied for it's own sake". I think there is not that much of philosophy of science as I take purpose synonymous with motivation Sep 2, 2022 at 13:01
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Depends on the perspective. From an individual perspective the motivation to study pretty much anything is/should be curiosity. You want to understand something and expand your understanding or the understanding of society as a whole. It can be tied to an immediate question or problem that you want to answer or solve or it could just be general curiosity, but you likely invest time because you're interested in the subject for a huge variety of possible reasons.

Or do you want to know why "we" as a collective whole "study", that is fund the research on mathematics, despite it's lack of application? Well it has historically proven to be very useful so that is still social capital that it can bank on and this "fringe nonsense" still occasionally happens to help out scientific domains in major ways. Not to mention that math is used in all of science so having a solid grasp on that gets you a long way in a lot of other domains so it's not necessarily wasted to have that skill set.

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It seems that utility is no longer the focus and I doubt many mathematicians would agree that the reason they study math is due to "aesthetics" as when one decides to be precise, there is not much more room for creative license.

On the contrary, mathematics is intensely utilitarian: all branches of mathematics are there to provide the tools to solve classes of problems, even if this isn't immediately intuitively apparent. The problem for the lay-person is that one of the main tools for the mathematician is the process of abstraction, which I sould suggest has a precise meaning in mathematics: for a given area of study, you construct an equivalence relation (or more generally an isomorphism), then study the structure that is left over after you 'throw away' the details of the objects that are equivalent in this sense.

As a concrete example, consider Set Theory, where you study general sets and the functions between them. Two sets are considered equivalent if there is a bijection between them - a bijection can be thought of as simply a 're-labeling', if you will. In this case, the structure that is left over is nothing more than the natural numbers - the only property that all equivalent sets have in common, is the number of elements. So, the set of natural numbers is an abstraction of set theory.

This may be a very abstract way of thinking about it, but nobody can deny that numbers are of great, practical importance.

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Math is the study of structure. There are plenty of structures around us that are simplified when put into math. A lot of people study mathematical structures purely for pleasure, without any idea nor thought about its applications. Sometimes, it has no applications... yet. Then, someone makes the connection.

Perhaps some mathematical structures have no (meta)physical analog of which the study of would be useful. We'll never know though, unless we figure out all of reality.

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