# What makes something mathematics?

What classifies something as math? Is "math" simply performing operations with a certain set of axioms in mind? Is "math" anything that involves numbers? What about mathematical logic?

Google definition of math: mathematics

Dictionary.com definition of math: The study of numbers, equations, functions, and geometric shapes (see geometry) and their relationships. Some branches of mathematics are characterized by use of strict proofs based on axioms. Some of its major subdivisions are arithmetic, algebra, geometry, and calculus.

Merriam-Webster definition of math: the science of numbers, quantities, and shapes and the relations between them

• with both definitions you gave, stuff like BNF, graphs, and the whole crap studied in discrete math subjects would not be math at all – Luis Masuelli Mar 14 '16 at 16:08
• Sometimes mathematics is trying to find a set of axioms, rather than working with a set of axioms ... – Mozibur Ullah Aug 14 '17 at 12:51

Both definitions are outdated. As Husserl put it already back in 1901:"Only if one is ignorant of the modern science of mathematics, particularly of formal mathematics, and measures it by standards of Euclid and Adam Riese, can one remain stuck in the common prejudice that the essence of mathematics lies in number and quantity". In antiquity mathematics was almost exclusively about numbers and simple shapes, over the course of middle ages and 17th century functions and equations joined them. In 19th and especially 20th century mathematics underwent another transformation, so its content came to be represented exclusively by sets, even relations and operations on sets are sets. In 1950s abstract structures par excellence, mathematical categories, were introduced and deployed in various subfields and applications.

Perhaps we are better off looking not at what changed but at what remained invariant. Here is a summary by MacLane, a founder of category theory:"Mathematics is the study of those structures which arise in different uses but with the same formal properties – and mathematicians aim to carry out that study by using proofs. This view, unlike Platonism, also accounts for the ways in which mathematics is used in other sciences". So to use Aristotle's terms, mathematics is a study of forms abstracted from matter. This is what makes it so broadly applicable, algebraic structure of a group is embodied in fundamental symmetries of nature and individual crystals, living organisms and works of art. Form can be expressed as numbers, shapes, symbols, sets, categories and as yet unknown higher abstractions.

Because abstract form is not subject to observation or experiment, which can only witness the concrete, mathematics coined a different truth standard which is its cornerstone and characteristic feature - proof. In its ideal limit it takes the form of deductive proof in axiomatic systems, where all results can be derived symbolically from a few explicit and formally expressed assumptions. However, while its beginnings go back to the ancient Greece this ideal was only fully formulated in the 20th century. It does not mean that proofs are expressed, or even ought to be expressed, as formal deductions, a mere possibility of such expression suffices. It also does not mean that mathematics is free of experimentation and heuristic reasoning, but their results are preliminary, and mathematicians always strive to validate them by proof in the end.

Reliance on proof as the standard of truth explains another of its characteristic features, demand for precision of definitions and arguments beyond that of natural language or even science, and the mathematical rigor. The rigor ensures that mathematics is logically transparent, precision and certainty of conclusions are exactly the the same as those of the assumptions, nothing is added in the middle. It also insures that it is highly sensitive to detail, no contradiction however minute is tolerated due to the law of explosion, and therefore highly testable. A 1993 article by Jaffe and Quinn about the nature of mathematics, its current state and relation to physics, and the role of proof and rigor in it, ignited a debate that offers great insight into modern perspectives. See the responses by several prominent mathematicians.

That mathematics has access to pure form without the benefit of senses has always been philosophically mystifying. It inspired Plato's theory of ideal forms, and Kant's theory of synthetic a priori, and Husserl's conception of formal sciences. Kant gives a deep, if somewhat dated, insight into how mathematics can be what it is:

"A new light broke upon the first person who demonstrated the isosceles triangle (whether he was called "Thales" or had some other name). For he found that what he had to do was not to trace what he saw in this figure, or even trace its mere concept, and read off, as it were, from the properties of the figure; but rather that he had to produce the latter from what he himself thought into the object and presented (through construction) according to a priori concepts, and that in order to know something securely a priori he had to ascribe to the thing nothing except what followed necessarily from what he himself had put into it in accordance with its concept".

And since every definition is a negation, as Spinoza put it, let us end with Husserl's delineation of the tasks of mathematics and philosophy:

"The construction of theories, the strict methodical solution of all formal problems will always remain the home domain of the mathematician.... If the development of all true theories falls in the mathematician's field, what is left over for philosophers? Here we must note that the mathematician is not really the pure theoretician, but only the ingenious technician, the constructor as it were, who looking merely at formal interconnections builds up his theory like a technical work of art... Mathematician constructs theories of numbers, quantities, syllogisms, manifolds, without ultimate insight into the essence of theory in general, and that of the concepts and laws which are its conditions... Philosopher inquires into the essence of theory and what makes theory as such possible".

• 1. regarding MacLane's characterization: in which concrete use does the whole structure of the integers arise? 2. If some Lewis-Carroll-like mathematician “abstracted” a crazy, badly inconsistent, unsalvageable axiom system from something, is this really still mathematics? Or rather… surrealist art? – wolf-revo-cats Sep 30 '17 at 4:30
• @wolf-revo-cats As I understand structuralists, those structures need not be instantiated in toto, concrete uses are rounded out into idealizations supported by formalisms. The proofs in those formalisms is what distinguishes mathematical "fiction" from fiction proper, its characters are far more strictly regimented. But the difference is a matter of degree. – Conifold Oct 2 '17 at 17:59

From a modern point of view mathematics is considered the science of formal structures. Simple examples of such structures are topological spaces, groups, vector spaces, differentiable manifolds.

A good overview of all fields of active mathematical research can be read off from the Mathematics Subject Classification, see http://www.ams.org/msc/msc2010.html

Number theory and geometry are two of these fields. Both are very old, but both have removed themselves very far from calculating with numbers or from studying plane triangles.

All mathematical theorems have been proved, based on clear-cut definitions and axioms. Otherwise a mathematical statement is considered a conjecture - waiting for either a proof or a counterexample.

Math is a word. As such, it is subject to the limits of linguistics. One such limit: its nigh impossible to get everyone to agree on its meaning. If I may quote the opening line of Wikipedia's page on "Definitions of Mathematics,"

Definitions of mathematics vary widely and different schools of thought, particularly in philosophy, have suggested radically different and controversial accounts.

Yes, the question you asked is so difficult that there is not just a section of Wikipedia's mathematics page dedicated to it, there is an entire page for it.

Personally, I find the things I call mathematics all have a common root structure: Model Theory and Proof Theory. Model Theory studies the semantics of mathematical models while Proof Theory studies the syntax of mathematical proofs. I have generally found that things I call mathematics boil down to one or both of these roots, while things I call not-mathematics do not.

• "Nigh" and "neigh" are somewhat different :) – hobbs Jul 24 '15 at 1:25
• @hobbs eep! quickly blames the spell-check program It couldn't be my fault, no! – Cort Ammon Jul 24 '15 at 1:27
• (I am sure horses cannot agree on the meaning of anything, either.) – jobermark Mar 14 '16 at 15:04

From an Intuitionist point of view, mathematics is the pared-down basic repository of human intuitions stripped of their actual reference to reality. (I see this as the clearer meaning behind Brouwer's somewhat abstruse analysis of mathematics as the experience of the single intuition of time. I think he went too far.)

If something has representations but not instances, it is mathematics. There are instances of accountants, of rabbits, or of love... Or of quarks, if only in terms of a pattern of observed behavior. Bella is an instance and not a representation of a Vampire: she 'exists' in a different way in some Kripkeesque 'world'.

There is no instance of a straight line -- you can't draw one, you couldn't see if you did, etc., even in ordinary fictions. There is no instance of a number -- the collection of five things is not the number five, which cares nothing for the things it enumerates. There is no instance of all the permutations of a collection, there are only physical groupings that might represent the individual permutations -- otherwise, the permutations of the positions of atoms and the permutations of the roots of polynomials could not really be the same in the way that they are treated by Polya theory and Galois theory respectively. There are only representations of these things.

Geometry, for instance, is mathematics because it originates in our functional intuitions about space, but it refines them to the point where we know that the lines and figures we are drawing are not what we are talking about. We are talking about something that idealises those useful experiences into something beyond the real, which leaves no trace on reality. No one can draw the mythical 'straight line', but we all relate to it from our normal view of spatial relations, sighting from one point or the other, seeing 'vented' rays come straight in a window, etc.

When we have a collection of five things, we say things about how that collection combines with other such collections that has nothing to do with the actual things, (and might be wrong on that account -- combine three foxes and four rabbits and, eventually, you have left at most four animals.)

So mathematics is the collection of shared human intuitions that can be detached from all instances and still retain meaning for us as an abstract pattern.

To my mind, defined as such, mathematics lies at an intersection between rational psychology, aesthetics, and logic. We know something meets the criterion of representing an intuition, if in a broad enough distribution of cases it precipitates the reaction of 'of course' or its stronger version 'aha!', so mathematics has an aesthetic base.

But what kinds of such discoveries retain meaning without the cause and instances is a psychological fact, not a fact of an external nature, so mathematics is properly a part of rational psychology.

And we found very early on that this kind of combination of absolutely pure intuitions are the only things that we seem to be able to truly faithfully process in our logic without introducing excessive complexity. So how these ideas can be combined, and the combinations can be communicated becomes the very basis of logic.

• "If something has representations but not instances, it is mathematics." +1 just for this; it should be bolded as it seems to be the core of your answer. :) – Wildcard Aug 29 '17 at 0:54

The mathematical definition of maths is it's the set of all possible self-consistent structures.

(Victoria Gould)

• What does it mean for a structure to be "self-consistent"? Are the integers self-consistent? Is the euclidean plane self-consistent? Is Minkowski space self-consistent? Is Zermelo-Fraenkel set theory self-consistent? What is an example of a structure that is not self-consistent? – WillO Oct 3 '15 at 22:39
• +1:This is actually formalism; what it does is allow an expansion of mathematical horizons; but it doesn't guide it's organic growth, it's movement and dynamics. – Mozibur Ullah Dec 25 '15 at 10:18
• @WillO "God exists since mathematics is consistent, and the Devil exists since we cannot prove it." (Andre Weil) – Drux Dec 29 '15 at 8:19
• @Drux: I asked what a "self-consistent structure" is. A random quotation containing the word "consistent" does not constitute an answer to that question. – WillO Dec 29 '15 at 17:28
• Was euclidean geometry a structure before Hilbert axioms? Was it mathematics? – Marco Disce Mar 13 '16 at 18:45

What is Art? Art is that which trained artists do, apparently said in all seriousness by the art-critic Danto.

Following this, and anecdotally, the physicist Isham when asked what is good mathematics, answered what good mathematicians do.

Circular?

True - but somehow coherent

• Dude, there are limits to relativism. In Erdos's world -- where at mathematician is a machine that turns coffee into theorems, drinking the coffee is still not mathematics. – jobermark Mar 14 '16 at 15:08
• @jobermark: I wasn't answering in all seriousness... – Mozibur Ullah Mar 15 '16 at 8:04
• So maybe the downvote is an overreaction, but I figured two points out of your 22k won't matter... – jobermark Mar 15 '16 at 16:12

I don't think there is a proper definition of "math". Some would say or have said it's the study of numbers and space (Algebra and Geometry).

Is "math" simply performing operations with a certain set of axioms in mind?

Well, I would say yes, although that might not be very illuminating as a definition. This view is called formalism. The idea was part of Hilbert's program.

Is "math" anything that involves numbers?

Yes, but not exclusively. Consider any kind of abstract algebra: group theory, ring theory, linear algebra, category theory. Numbers are indeed important, but abstract algebra tries to generalize concepts we know from numbers to other objects. Not to mention: geometry isn't per se about numbers either. The term "number" is questionable anyway; it mainly exists due to historical reasons. It is usually a synonym for natural number, integer, rational number, real number, complex number or maybe even surreal number. However, there is no agreement on what a number really is.

What about mathematical logic?

Mathematical logic is an area of math and math is an area of logic, according to Russell and Whitehead's logicism. I would agree. I would even say logic and math are the same thing, except most math that is done is very "high-level" logic.

Of course, math can be done extremely rigorously (à la proof-assistant-rigorously), rigorously and less rigorously (with pictures or convincing examples). Whether you call an informal argument "math" is probably up to you.

Math is the study of precise and useful thoughts. If you have a thought that you can make completely precise through some means (maybe axiomatizing it in some logical language) and it is useful in some well-defined sense, you're pretty safe calling it mathematics.

Mathematical logic is the collection precise of thoughts about precise thoughts, logic is the collection precise of thoughts about thoughts in general, analysis is the collection of precise thoughts about the real numbers and their subsets and mappings, category theory is the collection of precise thoughts about collections of objects and structure-preserving mappings between them, set theory is the collection of precise thoughts concerning our ability to collect objects together and apply predicates to them...

The process of making the thoughts precise and completely unambiguous (in some meaningful sense) is really the point at which they become 'mathematics'. Axiomatizing all of your assumptions and encoding the axioms in some commonly accepted logical language is often the most straightforward way to do this, where 'algebraic' properties emerge from the definition of symbols for binary/infinitary operations on individual 'elements' and their subsequent manipulation, 'topological' properties emerge from the definition of symbols for 'subsets' and their intersections/unions/sequencing, 'uniform' properties emerge from the definition of coverings ('sets of subsets') and refinement for your structure, so on and so forth.

If the thoughts are useful but can't be made entirely precise in some essential sense, they fall outside the domain of mathematics. If the thoughts are precise and useless, nobody tends to care. This has always seemed to be the defining nature of mathematics to me.

• Nice definition, but you slipped a bit when you said "completely precise." There is no such thing as "completely precise" in any universe, if you only deal with symbols. All symbols are subject to misunderstanding or alterations. A thought can be precise. It can even be completely precise. However, the symbol for the thought can never be "completely precise" except insofar as another mind agrees it to represent the same "completely precise" thought. So you are slipping from lack of attention to how minds relate to all of this. Still, +1. – Wildcard Aug 29 '17 at 0:59

There is a range of views among mathematicians and philosophers as to the exact scope and definition of mathematics. –Wiki

STEM is an acronym referring to the academic disciplines of Science[note 1], Technology, Engineering and Mathematics. –Wiki

If 'it' fits better in mathematics then it does in the other three, then it's (just basic, academic) math.

Mathematics is the study of common properties of things that have different physical content. That is study of properties which are common to physically unrelated things.

All triangles have common properties, be it triangle of atoms or triangle of the roof or triangle of stars. All sets of 6 items have common properties be it 6 apples, 6 cars or 6 galaxies.

People have noticed this and decided to study these properties separated from the physical things.

mathematics itself is only a particular formation of the mathematical.

In its formation the word "mathematical" stems from the Greek expression ta mathemata, which means what can be learned and thus, at the same time, what can be taught; manthanein means to learn, mathesis the teaching, and this in a twofold sense.

The mathemata are the things insofar as we take cognizance of them as what we already know them to be in advance, the body as the bodily, the plant-like of the plant, the animal-like of the animal, the thingness of the thing, and so on. This genuine learning is therefore an extremely peculiar taking, a taking where one who takes only takes what one basically already has. Teaching corresponds to this learning. [...] If the student only takes over something that is offered he does not learn. He comes to learn only when he experiences what he takes as something he himself really already has.

The mathematical is thus the fundamental presupposition of the knowledge of things

Heidegger, Martin. “Modern Science, Metaphysics, and Mathematics” in Basic Writings, ed. Krell. New York: HarperCollins, 2008. 271-305.

The mathematical is about teaching and learning ("in an originary sense") to make possible the knowledge of things.