# What is Mathematics?

#### Inquiring_Mike

What is mathematics? I like to think of mathematics as the language in which nature speaks... What is mathematics to you?

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#### Kerrie

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mathematics is description of the reality of the universe

#### sascha

mathematics is a language -- in contrast to others a thoroughly formalized one

what this language describes is another question.... sometimes people find out correlations with the external world only very much later than the formulation

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Only two replies and already a division of the house. Let me add another idea which doesn't square with either of the first two.

Mathematics is the study of patterns. This is a recursive study because the relationship of two patterns itself makes a pattern.

#### sascha

selfAdjoint, a division of the house can be inspiring. What you say -- "mathematics is the study of patterns" -- is closer to what I say than you may believe, because I had been thinking about posting that this language is about patterns of the mind (not describing them, but depicting / revealing them). Note that the recursiveness you mention is only formal, it does not cover content: the pattern resulting from relating two patterns is not the same pattern.

Something that bothers me more than that is one pattern that has been overgrowing and obscuring the rest: the gesture of algebraization, thinking in discontinuous terms -- as opposed to the geometrical way of thinking, in continuous terms. Geometry has been engulfed by algebraic analysis (while it makes quite a difference whether you understand e.g. a function as a curve or as a formula). Strangely enough, this difference is extremely rarely even discussed -- but has an enormous impact on the way thinking as such, as well as thinking about the world, are being understood.

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Sasha, in grad school I discovered that different mathematicians have different talents. It appears that the algebraic talent, doing things with abstract counters in your head, is more widely distributed than the geometric one, manipulating pictures in your head.

By which I don't mean that either algebra or geometry is those simple qualities, but that those talents are what you need to succeed in algebric or geometrical fields.

#### Kerrie

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Sasha, in grad school I discovered that different mathematicians have different talents. It appears that the algebraic talent, doing things with abstract counters in your head, is more widely distributed than the geometric one, manipulating pictures in your head.

By which I don't mean that either algebra or geometry is those simple qualities, but that those talents are what you need to succeed in algebric or geometrical fields.
so could the definition of mathematics be based on personal perspective of it? i see math as a description of our world around us, in the geometric sense, as basic geometry comes super easy to me...i suppose i can see why people call it a language-a universal one, but to me, language is a man made structure of communication rather...

#### sascha

By language I mean its very principle: being a structure of several signs, interrelated according to a basic idea (axiom, choice, desire, principle, etc.) -- as opposed to the very principle of thinking, which is to integrate and find the common basis of diversities. These two principles are not man-made; only this or that language, and this or that thought, are man-made.

Sure, a talent for this or that is not distributed evenly. But here we are seeking, if I understand this thread correctly, the systematic reason and structure of math, not some contingent aspects. Or not?

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Originally posted by sascha
By language I mean its very principle: being a structure of several signs, interrelated according to a basic idea (axiom, choice, desire, principle, etc.) -- as opposed to the very principle of thinking, which is to integrate and find the common basis of diversities. These two principles are not man-made; only this or that language, and this or that thought, are man-made.

Sure, a talent for this or that is not distributed evenly. But here we are seeking, if I understand this thread correctly, the systematic reason and structure of math, not some contingent aspects. Or not?
Well I see mathematics as a human enterprise, more or less inseparable from its sociology, like political science. A platonist would say the math is "out there" somewhere, but I think it exists in people's heads.

The reason it seems "out there" IMO, as I've posted before, is that the ideas that are accepted into the canon - published in journals and so on - have to be "well defined", meaning that their defining properties are clearly stated and related to each other (usually with an axiom system) so that it is no longer possible for informed people to disagree on their natures. This gives them a mental "solidity" similar to that of perceived rocks and chairs.

#### FZ+

I veer between Platonic:

Mathematics is the language in which the laws of the universe is written.

And formalism:

Mathematics is the language in which the laws of the universe is read.

#### sascha

Maybe the point is less an 'either–or' (of e.g. the Platonist vs. formalist approach), but of realizing that the aspects and interconnections between laws and their appearances on all levels are precisely a question of how we approach totality, 'the whole thing'. I advocate thus an 'as-well-as' overall view. This does not mean to relativize blindly, but to relate any (assumptional, presuppositional) approach to its (logical, predicative) effect and gradually finding an ever more complete grasp of grasping -- i.e. not only seeing the laws or their effect, the 'writing' or the 'reading', but both. AFAICS, there are presently four types of approach to the philosophy of mathematics, of which none is strictly conclusive: Platonist, conceptualist, formalist, and intuitionist. My aim is more to understand the reasons (relation between assumptions / presuppositions and their logical effect) behind these approaches, rather than to attach myself to one of them.

#### Dal

For some kids, Mathematics is a nightmare.

#### sascha

Mathematics is a nightmare only when taught in ways that leave no space for the creative quests that arise when hitting upon the structures of well organized thinking. Of course, in an epoch like ours -- in which everything is being formalized and instrumentalized, making it lifeless, sacrificing the overview for getting lost in details -- math can indeed be made into a nightmare. I remember how reading George Polya aroused my interest after having felt for a long time that math is a real drag. In France there was an interesting book "Echec et maths" (a double word game: failure / chess and checkmated / maths), showing how kids are pushed into falling though, as an effect of the mentioned 'pedagogical' ideas.

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#### Dal

I was unlucky enough to have been taught in a system that brings nightmare (oh yes the real ones). My first school was in Geneva and when I was about 7 or 8, I went back to Burma. Then comes the shock: everything turned upside down from the moment I first entered the school. I hated school ever since then, and especially math. But luckily the story didn't end there, and gradually I came to love physics and math.

To get back to the topic; I imagine mathematics is a language that describes what our oral languages lack to describe. Quantitively. But far from being universal because I believe there are many flaws in it. Mathematics doesn't describe the principles and laws but merely a language that we use to understand the description of the universe. If mathematics is a language universal and can describe every principles in the universe, we would have found the grand unified theory itself.

#### sascha

There are probably only contingent pragmatic reasons why mathematics is usually subdivided into a "pure" and an "applied" sector. The question then is, in which one does one situate oneself, maybe even forgetting about the other one. This might concern your characterization of mathematics as something quantitative. This may be reasonable to some extent for applied math, but at least pure mathematics ought to be more open than that, otherwise it will gradually self-limit itself, become one-eyed, etc..

In this problem, I would like to mention a point that Hegel has demonstrated quite clearly and which should not be forgotten: All quantities require, for being formulated, a qualitative decison. Any quantity implies a qualitative foundation. It cannot be defined without that. Of course one can forget about this foundation. But that is quite another question. Then one will face the consequences in inherent limits of the resulting systems -- which is especially fateful when operating in a discipline which should be free of biases for remaining on track.

It is interesting that in a former thread here on mathematics (now in the Archive), the participants had more or less agreed on the idea that mathematics is based on the numbers -- and then were compelled to argue endlessly about the resulting problems. "It is all in the head", the problems are all self-made.....

#### phoenixthoth

different mathematicians will give different definitions.

here's one way:
mathematics is a branch of knowledge characterized by the following:
1. rules of logic and deduction are studied and/or adopted.
2. what is considered a proof is studied and/or agreed upon.
3. terms are used with or without having a definition. the words without definitions are kept to a minimum but that they are undefined remains recognized.
4. assumptions are studied. some are agreed upon.
5. theorems are written which combine steps 1, 2, 3, and 4.
6. consequences of theorems (using 1-4) are written.
7. attempts at generalizing the theorem are made.
8. conjectures are studied; one looks for them to be proven as either a theorem, not a theorem, or undecideable.

within this greater definition, you can have a wild variety of the particulars when it comes to steps 3 and 4.

sometimes, a particular case of mathematics will in some way resemble reality but that is NOT the defining aspect of mathematics unless by "reality" you mean more than the physical/observable. keep in mind such fields as metamathematics and category theory.

whenever one attempts to define mathematics, it is often the case that a field within mathematics would be left out by that definition. the average definition would not include category theory, metamathematics, and arithmetic; i believe that this definition leaves no field out.

however, in its generality, other things not commonly considered mathematics are drawn in such as philosophy and science, but this appears to be a part of a larger suspicion that many branches of knowledge resemble each other at the abstract level. one can probably invent a similar definition of philosophy and then consider mathematics a branch of it. once, everything was considered a branch of philosophy.

cheers,
phoenix

#### Jug

What is Mathematics

Mathematics is the science of describing exacting relationships.

"All things number and harmony." - Pythagoras

#### NateTG

Homework Helper
Mathematics is the study of patterns. This is a recursive study because the relationship of two patterns itself makes a pattern.
I like to think of it as the study of abstraction, but the notions are essentially the same.

#### phoenixthoth

just out of curiosity, what is the point in defining terms like mathematics, art, philosophy...?

#### sascha

Phoenixthoth, why does for you asking a question about the nature of a field or discipline amount to defining a term?

#### phoenixthoth

it doesn't. in fact, one could say that's what i was getting at, that defining it is pointless; discuss its nature.

but this thread is titled "what is mathematics" not "what is the nature of mathematics?"

having said that, i think my definition of math captures some of its nature.

#### sascha

ah, interesting. I was thinking that the point of asking the question "what is mathematics?" is to debate the nature of mathematics; otherwise one can collect definitions anywhere. On the other hand, since in your own opinion defining mathematics is pointless, and that we should discuss its nature, why do you yourself give a definition (your post of 6 October)?

By the way, I do think your definition leaves out some rather essential features. Think of the basis of projective geometry ("Geometrie der Lage", Felix Klein and others) that contains all geometries and sets out on a purely qualitative path (i.e. with no formal definitions, only the intuitive idea on the point, the line, the plane, in space). By thinking infinity rigorously at every step, it gradually develops all that is needed, including the concept of counter-space, or imaginary points. In this geometry, infinity is never a special case. And numbers enter the scene in a natural way, not as with set theory, where you always have the problem whether the (postulated) entity called "set" is an entity of the same sort as the others which one wants to deal with. The result of the problem is Russell's antinomy.

The projective geometry line of mathematics has difficulties within your definition, which seems in other words a bit too narrow. But by definition, definitions are always too narrow, because they nail down the subject matter before it is explored. In this context: What do you think of my post of 29 September?

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#### phoenixthoth

which 9/29 post?

i'll guess this one:
mathematics is a language -- in contrast to others a thoroughly formalized one

what this language describes is another question.... sometimes people find out correlations with the external world only very much later than the formulation
i looked at the definition of the word language on dictionary.com and mathematics seems to be a language.

however, to me, this is like defining a chair by saying "a chair is an object." in other words, yes it is a language but just saying that leaves about as much to the imagination as "a chair is an object" leaves about a chair.

whether or not math is more formal than say, english or latin, is disputable. when i look at a grammar book, it all looks pretty formal with all its rules to me. but this isn't really a central issue. i suppose the "grammatically correct" statements in mathese would be theorems and proofs.

what is the point? to kill time i guess.

i'm not seeing how projective geometry is not in my definition of math. "with no formal definitions, only the intuitive idea on the point, the line, the plane, in space." in my number 3, i said that some terms are not defined. one could call this lack of definitions an intuitive/qualitative domain. in fact, geometry and set theory were two things i had in mind when i said that not everything has a definition, eg sets and points. if you use the word "infinity" then you're doing a special case of #3 in my definition.

the projective geometry i'm familiar with is not at all just qualitative and it has formal defnitions.

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#### sascha

Sorry for being so slow in answering; I had some technical problems.

In fact I meant my post after the one you quoted, the one after selfAdjoint. There I say "Geometry has been engulfed by algebraic analysis", which corresponds to what you tell me about the projective geometry you are familiar with. This is just my point: the universality of the subject matter (as possible in a purely qualitative approach, thinking in geometrical way of thinking, in continuous terms) has been lost by subjecting it to the gesture of algebraization, thinking in discontinuous terms. This is the fashion nowadays, and mathematicians have not yet grasped what they have lost. As little as mainstream philosophers, by the way, who also resorted to thinking in discontinuous terms.

My saying mathematics is a language is not meant as a definition, but as a clarification for taking off some of the false gloss which some put onto it, believing mathematics is more than a language (e.g. something that warrants per se correctness in dealing with material reality). It is possible that #3 in your definition covers intuitively what I mean. But I think the distinction of continuous versus discontinuous approach still needs to be more explicit for expressing fully what I mean. This is indeed not discussed very often. The last I remember is an article by the Bourbaki group, many decades back. You are right in your post of 6 October that everything deeply considered becomes a branch of philosophy. But this is valid even today.

#### phoenixthoth

There I say "Geometry has been engulfed by algebraic analysis", which corresponds to what you tell me about the projective geometry you are familiar with. This is just my point: the universality of the subject matter (as possible in a purely qualitative approach, thinking in geometrical way of thinking, in continuous terms) has been lost by subjecting it to the gesture of algebraization, thinking in discontinuous terms. This is the fashion nowadays, and mathematicians have not yet grasped what they have lost.
what do you mean by contiuous?

i don't think mathematicians are not "thinking in geometrical way," in general, when they think about projective geometry. it's not like they state a bunch of equations without ever refering to geometric objects. and i'll bet that the geometric way of thinking inpsires the direction the algebra is taken in. i'll also bet that directions in algebra can give new directions in the geometry. i think to totally ignore either side of it would be to lose something essential. i'm not seeing what's wrong with codifying geometric intuitions in algebraic form.

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