What makes something mathematics?

Question

Dictionary.com definition of math:

(used with a singular verb) the systematic treatment of magnitude, relationships between figures and forms, and relations between quantities expressed symbolically. (used with a singular or plural verb) mathematical procedures, operations, or properties.

Merriam-Webster definition of math:

1 : the science of numbers and their operations (see operation sense 5), interrelations, combinations, generalizations, and abstractions and of space (see space entry 1 sense 7) configurations and their structure, measurement, transformations, and generalizations Algebra, arithmetic, calculus, geometry, and trigonometry are branches of mathematics. 2 : a branch of, operation in, or use of mathematics the mathematics of physical chemistry

These are general definitions, but certainly philosophers have more to say about what mathematics is. 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? What makes something mathematics?

Answer 1

Both definitions are outdated.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".

Answer 2

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.

Answer 3

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.

Answer 4

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.

Answer 5

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.

Answer 6

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.

Answer 7

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.

Answer 8

Short Answer

'What is mathematical' is one of the primary concerns of the philosophy of math, and depending on one's beliefs about and in philosophy, particularly one's metaphysics, there are very divergent views on what it means to be mathematical. An approach to defining 'mathematical' (besides as a synonym for 'precise') might fall under two approaches, one a general, natural language definition that might conform to prototype theory and one that follows more technical questions of intension that draws on questions of necessity and sufficiency.

A general response to 'mathematical' would follow the common-sense definition that would invoke terms such as quantity, qualities, relations, shapes, operations, truth, and direction and would appeal to some common systems of mathematics like arithmetics, algebras, vectors, and geometries. The technical response to what is mathematics and where it comes from would be more in line with the various schools in the philosophy of mathematics. These responses are far more technical but include ideas like reduction to logic, formal systems, intuition, empirical, abstraction, iteration, and semantics. The former class of characterization is comprehensible to most people, whereas the latter class is technical philosophy drawing particularly on the jargon of epistemology and ontology.

If what is 'mathematical' were voted upon by the set of educated adults, the former class would be the most popular, but whether or not one finds that appealing would be related to one's views on the use of ordinary language in philosophy and the sophistication of one's philosophical jargon. It should be said, that any claims to certainty about what is 'mathematical' would conflict with any epistemological approach that is pluralisitic and fallibilistic (IEP). The claim that Pythagoreans threw competitors off a cliff (Reddit) is likely to be apocryphal, but it certainly is believable.

Long Answer

The Politics of Mathematics

Any answer to what is mathematical is a lot like an intellectual fingerprint. Take for instance the mathematical philosophy of Wittgenstein, it differs remarkably from the claims about 'mathematical' made by Aristotle (accounting for the fact that mathema, 'μάθημα' in Greek languages is something much broader than what we call mathematics).

To demonstrate how the definition of 'mathematical' is highly partisan and divergent, let us consider the characterization of the 'mathematical' from an empirical perspective and what the SEP has to say:

he general philosophical and scientific outlook in the nineteenth century tended toward the empirical: platonistic aspects of rationalistic theories of mathematics were rapidly losing support. Especially the once highly praised faculty of rational intuition of ideas was regarded with suspicion. Thus it became a challenge to formulate a philosophical theory of mathematics that was free of platonistic elements. In the first decades of the twentieth century, three non-platonistic accounts of mathematics were developed: logicism, formalism, and intuitionism. There emerged in the beginning of the twentieth century also a fourth program: predicativism. Due to contingent historical circumstances, its true potential was not brought out until the 1960s. However it deserves a place beside the three traditional schools that are discussed in most standard contemporary introductions to philosophy of mathematics, such as (Shapiro 2000) and (Linnebo 2017).

If one takes the position that mathematical thought (whatever it may be) reduces to psychological theory such as recognized since the psychologism of the 19th century, then one might align with John Stuart Mill's mathematical philosophy (SEP) as an inductivist endeavor:

Amongst the Laws of Nature learnt by way of inductive reasoning are the laws of geometry and arithmetic. It is worth emphasizing that in no case does Mill think that the ultimately inductive nature of the sciences—whether physical, mathematical, or social—precludes the deductive organization and practice of the science (Ryan 1987: 3–20). Manifestly, we do work through many inferences in deductive terms—and this is nowhere clearer than in the case of mathematics. Mill’s claim is simply that any premise or non-verbal inference can only be as strong as the inductive justification that supports it.

This approach to understanding 'mathematical' may not be as popular as Husserl's, it rejects transcendentalist notions derived from Kant that an eliminative materialist would be okay with. Linnebo in his Philosophy of Mathematics has a chapter on contemporary empirical interpretations of 'mathematical'.

Empirical, Intuitional, and Unorthodox Accounts

Linnebo raises the a point about the relationship between the empiricism of Locke and mathematical thinking on page 88:

[T]he very idea that mathematical knowledge is synthetic a priori clashes with the empiricist creed that all substantive knowledge is empirical.

He goes on to cover Mill and Quine's contributions.

In Chapter 8: Mathematical Intuitionism, he suggests that intuitionalism is empirical in nature but rejects the excesses of empiricism on page 116:

We shall now take a closer look at some accounts of mathematical knowledge that do not assimilate it to empirical knowledge... [since] some form of mathematical intuition provides evidence for certain mathematical truth.

One unorthodox approach to characterizing mathematics comes from George Lakoff and Rafael E. Núñez in their Where Mathematics Comes From. Broadly speaking, the answer to the title of the book is psychological processes of the mind that can be understood as conceptual metaphors. One example they give in their book is the relationship between the minds fundamental linguistic capacity to categorize and using the Metaphor of Containment as understanding the intuitive sets of naive set theory. It's an obvious empirical truth that people understand containers long before they develop formal mechanisms for defintions of extension. Of course, while the idea that the brain is strongly related to the mind through neural correlations of consciousness, there is still a lot of woo in mathematical philosophy (thus revealing my bias).

Summary and Advice

If you're looking to understand what is 'mathematical' and what is not, like most fields in including attempts to define 'science', there can be friction over demarcation and definition. If you're looking to understand what is 'mathematical' and you don't have some experience in undergraduate math, it might help if you learn a little non-Euclidian geometry, abstract algebra, and formal set theory before wading into the very abstract and technical notions of mathematical philosophy which has a very fascinating explosion of ideas since Frege Gottlob invented the basis for the formalisms used today in mathematical logic. At the bare minimum, as you develop your own beliefs, you'll be able to understand some of the basic theories that underlie technical notions of mathematics and begin to see as Thomas Kuhn did of science, that mathematics is conducted by mathematicians, many of whom get paid to defend their beliefs or have professional reputations (not to mention feelings) at stake when they peddle philosophies that may largely reject contemporary science and persuade because of the practical outcomes of staying on the bandwagon.

Answer 9

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

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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.

Answer 10

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.