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Philosophers have given the nature of mathematics a lot of thought. As a beginner exploring philosophy, one of the questions which presents itself is 'what is X', and in this case, X is mathematics.

Mathematics is almost universally taught and used, at least at the arithmetic level, and while someone can study more sophisticated math, it seems one can start asking fundamental questions about what doing arithmetic, or even more generally mathematics is about.

Instead of 'What is mathematics?', let me ask, what are the predominant, contemporary schools on 'what mathematics is'? What are some immediate references that I can use to explore these positions on the nature mathematics?

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    One line questions are discouraged, and philosophers "agree" on hardly anything.
    – Conifold
    Nov 4 '21 at 21:51
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    Broad edit to fill an introductory-level question that is not answered in our knowledge base.
    – J D
    Nov 5 '21 at 15:26
  • Mathematics is the study/science of measurement. That's what it boils down to, anyway. Just sayin'... Nov 5 '21 at 18:10
  • @TedWrigley: That is no good for the breadth of what mathematics is now, & I'm dubious about it describing origins - for instance zero & infinity are said to have originated in Buddhist & Jain religious thought, in relation to spiritual exercises/thought experiments. See bbc.co.uk/programmes/p0038xb0
    – CriglCragl
    Nov 6 '21 at 1:07
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    @CriglCragl: Having none of something is the first truly abstract measurement. Zero isn't spiritual so much as it's metaphysical: the existence of non-existence. That's why it has appeal in certain non-Western philosophies Nov 6 '21 at 5:42
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Short Answer

In philosophy, there are questions that have pat answers, such as what was Kant's primary language, and then there are questions that do nothing but raise more questions. 'What is mathematics' is of the latter. The philosophy of mathematics addresses this/these questions, and provides as a starting ground for additional reflection. There is the general definition of 'what mathematics is' according to MW:

Definition of mathematics

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

But philosophical definition might be better be understood as the art of characterization. Three common ways mathematics is understood are through logicism, intuitionism, and formalism. If you develop a basic understanding, you're on your path to having your own answer on 'what mathematics is'.

Long Answer

What is mathematics is a tremendously broad question by today's standards of philosophical discourse. But it is an absolutely recurring question for those who are starting to reflect on metaphysical principles. A coherent philosophical response can occur on different levels. If you have a background in category theory and are familiar with the history of Hilbert's program, it may take one form. I'm going to presume you're not a graduate student in mathematics and answer this as if you are just becoming interested in the philosophy of mathematics. Certainly, not everyone who is interested in the philosophy of math can calculate tensors and eigenvectors.

This broad question places you firmly in the domain of encyclopedia entries. Start with Philosophy of Mathematics (SEP).

A quick summary from this article suggests four contemporary schools:

2.1 Logicism
2.2 Intuitionism
2.3 Formalism
2.4 Predicativism

Let's take a look at the first three!

Logicism

Logicism is very popular and sprung from the work of Gottlob Frege who is some consider one of the fathers of the linguistic turn in philosophy. Although relatively unknown, Bertrand Russell brought attention to his work and championed it alongside some very famous philosophers such as Alfred North Whitehead. They may have written a work about it.

From the article:

the philosophy of mathematics, logicism is a programme comprising one or more of the theses that — for some coherent meaning of 'logic' — mathematics is an extension of logic, some or all of mathematics is reducible to logic, or some or all of mathematics may be modelled in logic.3 Bertrand Russell and Alfred North Whitehead championed this programme, initiated by Gottlob Frege and subsequently developed by Richard Dedekind and Giuseppe Peano.

For a taste of this philosophy of mathematics which is largely a reduction of mathematics to formal logic, consider the logicist conception of the number system and arithmetic encompassed by Peano's Axioms (PA), one of the elemental achievements in mathematical logic. From Wolfram:

  1. Zero is a number.
  2. If a is a number, the successor of a is a number.
  3. zero is not the successor of a number.
  4. Two numbers of which the successors are equal are themselves equal.
  5. (induction axiom.) If a set S of numbers contains zero and also the successor of every number in S, then every number is in S.

Doesn't seem very 'mathy', does it?

Intuitionism

Intuitionism is a school which rejects the notion that mathematical objects are somehow 'out there', and accepts that math is an activity of the human mind:

In the philosophy of mathematics, intuitionism, or neointuitionism (opposed to preintuitionism), is an approach where mathematics is considered to be purely the result of the constructive mental activity of humans rather than the discovery of fundamental principles claimed to exist in an objective reality. That is, logic and mathematics are not considered analytic activities wherein deep properties of objective reality are revealed and applied, but are instead considered the application of internally consistent methods used to realize more complex mental constructs, regardless of their possible independent existence in an objective reality.

Given our understanding of the working of the brain, and married with our knowledge about computers, it might be hard to see how philosophers might reject this, however, a longstanding understanding of mathematics was a product of Platonism which might be oversimplified as the belief that in some sense, circles are 'real' objects that exist in some alternative world of being that we have access to through our thoughts. Just as a modern-day Christian might believe in an actual place called Heaven populated by actual beings called Angels, a platonic thinker might believe there is a Real of Forms inhabited by real objects called Circles. As a naturalist who rejects supernaturalism, I find both ideas distasteful, but that doesn't mean my beliefs are true. These are fundamental discussion in metaphysics related to ontology and epistemology.

Formalism

If you are a computer scientist, formalism may be the approach for you! With its emphasis on defining mathematics as operations on formal systems, there is a certain appeal to seeing mathematics as visual language games with complex rules, like the lambda calculus.

(I'd cite some typical fare from a highly abstruse formal representation, but PhilSE doesn't do math formatting well.)

This approach emphasizes how mathematics can be shown to be computable. David Hilbert himself was a proponent of this aspect of mathematics given his love of formal systems.

What Is Mathematics and Where it Comes From

These are broad questions, but after working your way through some encyclopedic entries, you might want to start with some primary sources. A classic text is Bertrand Russell's Philosophy of Mathematics. There are contemporary authors like Oystein Linnebo's Philosophy of Mathematics which covers other lines of thought that intersect with mathematics such as nominalism. Ultimately, though, be prepared to learn some mathematics along the way, since the logic of mathematical discovery often feeds the logic of mathematical philosophy.

There is no simple answer to defining mathematics, and like any philosophy, questions tend to grow faster than answers. But if you ask enough questions, eventually you'll find some satisfaction in the collection of answers.

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    You've strawmaned platonism. So here's the antiplatonist strawman youtu.be/Zh3Yz3PiXZw for you. In short if you believe there are "right answers" out there you're platonist. Nov 7 '21 at 5:51
  • @Rusi-packing-up "Out there" since empiricism is the atmosphere or space. If you believe in mind-independent algebraic equations floating in space, then you've missed the growth of knowledge since Plato. Plato wouldn't peddle neoPlatonic thinking were he alive. He'd get Russell's teapot argument well enough. All cognition is embodied until "out there" can be demonstrated continuous with physical casualty. No brains, no minds, unless you can demonstrate otherwise. Loved the video! : ) Btw, I'm open to a compeling argument or demonstration that this "out there" exists!
    – J D
    Nov 7 '21 at 6:40
  • @Rusi-packing-up I reflected on your use '"out there"' with it's quotation mark is a fascinating thing. On the one handit purports to be objectively true because the phrase 'out there' is idiom for the objective and publically accesible, but on the other hand, it strips the phrase of that meaning by invoking double quotes to announce that '"out there" isn't really the same 'out there' that is the received definition'. That means your "out there" has no reference. Onus probandi requires you to then you to provide the basis of your ontological commitment to this alternative "out there"...
    – J D
    Nov 7 '21 at 14:15
  • It's not an extension of space-time, fine. But "out there" also rejects seemingly that is mental experience. Because one can claim there is something "out there" that is neither mental nor physical doesn't instantly unseat the intuitions nor as far as I can tell the justifications of the competing claim there is only the mental and physical.
    – J D
    Nov 7 '21 at 14:18
  • This would seem to be a claim about some transcendental object of thought or being, if we set aside mental and physical implications on those terms, resp.How am I supposed to justify through rational and/or empirical means that which is neither of mind or body? What other options are there beside faith?
    – J D
    Nov 7 '21 at 14:21

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