"There's no sense in being precise when you don't even know what you're talking about." (John von Neumann)

Prologue

For the typical primary school or high school student, the following definition of mathematics would suffice:

(Naive definition) Starting from obviously true axioms, use obviously correct inference rules to derive additional truths.

In this sense, mathematics is all about discovering indisputable truths. For example, the theorems proved in geometry would be literally true statements about the physical space. Someone may argue that there is no such physical object as a geometrical point, or a geometrical line, but this is no issue because we can reply that geometrical objects are nothing more than locations in the physical space, and thus they can happily exist even if nobody can see them materialized. As for the exotic topic of complex numbers, they can be viewed as a man-made tool that sometimes comes in handy for mathematicians in describing reality.

Historically, this looked plausible until the early nineteenth century, when non-Euclidean geometries were invented. It wasn't generally believed any longer that Euclidean geometry was the true description of physical space. Curiously, it turned out to be impossible to decide which geometry, if any, was the true one. But then it raises the question: if mathematics does not necessarily describe reality, what does it describe then? After some meditation, we may adjust our previous definition as follows:

(Modern definition) Starting from obviously clear assumptions, use obviously correct inference rules to derive consequences.

Here we acknowledge that mathematics is all about discovering the logical consequences of given assumptions, where it is not required that the assumptions are actually true. It's comparable to making logical deductions based on the "facts" set out in a detective fiction.

Again, this looks plausible until we find out that different groups of mathematicians don't even agree on which logical rules should be permitted when deriving consequences. Most notably, certain groups don't accept the unconditional application of the law of excluded middle. This stems from their divergent views on the existence of mathematical objects.

What was told so far shows that defining mathematics is far more involved than expected. And we didn't even try to define the scope of mathematics, only its methodology was considered. To move things forward, the attempt to define mathematics must be accompanied by a better understanding of existing approaches to the foundations of mathematics.

One remark before we continue: in the definitions given above, "obviously true", "obviously correct", and "obviously clear" do not mean true, correct, and clear, respectively. Such a wording was used solely to indicate things that, for most people out there with a brain, would or used to appear as obvious. It is this entanglement with the obvious what makes mathematics possess the illusion of indisputability.

Philosophy of mathematics

Philosophy offers reasonable arguments about topics where, given our current level of knowledge, there is no feasible way of testing or verifying any theory. The foundations of mathematics is a philosophical topic of active research. This suggests that today we are still far away from a definitive answer as to what mathematics really is.

Generic framework

(Metaphorical definition) Mathematics is the intellectual discovery of nature's eternal, immutable infrastructure.

By "infrastructure", I mean that the discovery ultimately targets fundamental, ubiquitous ideas that we sense when observing appropriate configurations of things. (I am deliberately imprecise here, in the spirit of the quotation under the title of this essay.) By "intellectual", I mean that the process of discovering happens entirely within one's mind.

All ideas in mathematics are tied to observations. For example, when we look at {apple, apple, apple}, or {orange, orange, orange}, we sense "threeness", denoted by the symbol 3. When we observe the edge of a ruler, we sense a "straight line segment", even in spite of knowing that it would not look smooth through a magnifying glass. Looking at a computer network plan, we sense the idea of a "graph". Imagining a row of natural numbers, starting from 0 and fading away in the distance, can lead us to sense what we'd call the "set of natural numbers", denoted by ℕ. When we throw the dice, we sense "randomness". When we think of all humans ever born, we sense "potential infinity" (the actual set is finite at any given point in time, it's never completed, and may grow indefinitely). When we write down √-1, we sense something weird as if a number whose square is -1 existed (it sounded really weird before complex numbers became widely accepted). When we ask if there may be a positive quantity smaller than any positive real number, we sense the "infinitesimally small".

The ideas in the previous paragraph are sensed with varying vagueness. While "threeness" is pretty clear, √-1 and "infinitesimally small" feel like guessing. Since all the ideas are unclear to some (varying) extent, we need to invent theories in order to describe them and their relationships.^{(1) (2)}

As for the methodology, the starting point of mathematical theories are fundamental, ubiquitous ideas (incl. logic) about which we have accumulated so much and so consistent day-to-day experience that makes it possible to confidently rely on our intuition. We just close our eyes, and in an iterative process, come up with new ideas based on the already available ones, think out or take note of assumptions about the ideas we have, and discover the logical consequences of the assumptions.⁽³⁾ The only constraint is that the resulting theories must constitute conceivable stories about our world. As a minimum, statements including "clear" ideas (e.g. natural numbers, finite sets) should coincide with our experience, and statements including "unclear" ideas (e.g. √-1, infinitesimals, infinite sets) should not lead to known contradictions.^{(4) (5)}

The reason for mathematics transcending cultures and millennia is that humans have always had very similar experiences about the fundamental, ubiquitous ideas from which mathematics emerged. An alien civilization, if any, might develop a mathematics very different from ours, provided they exist in a very different environment and/or have very different sense organs.

The merits of a mathematical theory are assessed based on its beauty, success, and consistency. With regard to the development of mathematics, the first aspect is the most important guiding principle. The reason is, as once somebody put it, that beauty is felt as a result of sensing a deep law of nature. (I am deliberately imprecise here too.) As opposed to other sciences, the aim in mathematics is thus not to pinpoint and test, but rather to reflect laws of nature. That is, mathematics is both a science and an art.

In the following, standpoints of various philosophical schools are presented and commented. They can be used to customize the framework outlined above. It's a matter of personal preference.

Fictionalism

According to fictionalism, mathematics is a collection of useful fictions whose statements are, despite their usefulness, actually all false. In these fictions there are recurring "characters" like numbers, straight lines, graphs and many others, all entirely fictitious. Nevertheless, the fictions are useful because they convey (or rather, reflect) truths about our world. Furthermore, discussing our experiences in terms of carefully chosen, representative fictional characters greatly facilitates communication.

Although I agree that mathematics is a collection of stories, I still think that the ideas (i.e. the characters) in those stories are real, in one way or the other, simply because we do sense them. The assumptions the stories make about the ideas, however, may well (all) be fictional. It's like writing a guide about an existing city without knowing it well.

Also, in my view the ideas exist right here with us (just like the city in the previous analogy), not only in a separate "world of ideas" as platonism would suggest. E.g. in a computer network, there is a graph right there belonging to the network; where else could it be? Putting it another way, I don't think the network is more real than the graph.

Constructivism

Loosely speaking, constructivism means seeing is believing. The principle is that only those ideas and properties exist for which we can exhibit an appropriate configuration in terms of an agreed way of representation. For example, if it's agreed to represent real numbers via Cauchy sequences of rational numbers, then the square root of 2 exists only after one has constructed an appropriate Cauchy sequence. Before that, the square root of 2 does not exist, however esoteric this may sound.

The allowed ways of construction differ in different flavors of constructivism. In most cases though, the set of natural numbers is either assumed to exist or allowed to be constructed, either as actual (completed) infinity or as potential (incomplete) infinity. Moreover, instead of carrying out a construction (e.g. that of a square root), it may be agreed that it suffices just to provide a feasible method for the same.

Truth values have to be constructed too. Here the "appropriate configuration" is the concrete proof, and the "agreed way of representation" is the allowed forms of proof. A statement does not have a truth value until it has either been proved or disproved (or until a feasible method has been provided that would certainly result in a proof of truth or falsity).⁽⁶⁾ If there is no such truth value to observe, it simply does not exist for a constructivist, leaving the statement undecided.⁽⁷⁾

To prove "A or B", we need to prove at least one of them. This requirement lies at the very heart of constructivism, and follows from a "seeing is believing" interpretation of logical disjunction. Accordingly, to be able to say that "A or (not A)" is true, we need to either prove A or disprove A. This is different from classical mathematics where "A or (not A)" is always true in itself, since there it is taken for granted that every statement has a truth value, even if nobody can actually observe it. Similar applies to "not (not A) ⇒ A": in constructivism, the left hand side only means we've demonstrated that there is no way of disproving A; but it does not necessarily imply that A is true, or that A has a truth value at all.

Adhering to the principle of constructivism lends constructive mathematics certainty and confidence, and leaves little room for unpleasant surprises like paradoxes or contradictions. Eventually, it's hard to imagine a more obvious and tangible evidence of existence than that of a constructed representation. The price we pay is that proofs tend to become unusually cumbersome.⁽⁸⁾

Strict finitism

Infinity in mathematical theories has always been a major source of controversy. We can only perceive a finite thing in its entirety, and this is true for our imagination too. With respect to intuition, this means we can only guess what a real infinity would look like in terms of our finite perception, and whether there exists infinity of any kind (actual or potential) at all. To eliminate this guesswork, theories in strict finitism are free from infinity.

In classical mathematics, one can state that the formula n²-1=(n+1)·(n-1) is true for all natural numbers. In strict finitism, the corresponding statement is that any concrete equation that matches the above formula is true. In other words, we only state that if someone gets hold of a concrete natural number, say, 19, then the resulting concrete equation, 19²-1=(19+1)·(19-1), will be true. That is, while in classical mathematics the statement refers to all the elements of a (hypothetically) existing infinite set, in strict finitism it's merely an abstraction of concrete individual occurrences.

Infinite sets of classical mathematics may have counterparts in strict finitism. For example, the "set" of even numbers is basically defined as the property of being even. If a concrete number has this property, we say that the number is an "element of the set". So it's not really a set in the classical sense, but rather a common property that ties concrete occurrences together.

A "sequence" of rational numbers can be defined as a (finite) method that expects a single input, a natural number, and is guaranteed to produce a rational number as output in finitely many steps. Such methods can then be used to represent real numbers by finitary means.

Strict finitism is usually coupled with constructivism (constructivism taken to the next level, so to speak), but nevertheless it's possible to develop non-constructive theories that do not make use of infinity.⁽⁹⁾

Pure vs. applied mathematics

Pure mathematics deals with discovering about the ideas we sense, while applied mathematics means modeling real-world phenomena using a mathematical theory.

As an example, developing (the story of) Euclidean geometry, i.e. intellectually discovering the properties of and relationships between ideas like points, straight lines and planes, is pure mathematics. On the other hand, modeling shapes and trajectories of physical objects by means of Euclidean geometry, with the aim of making measurable predictions about them, is applied mathematics.⁽¹⁰⁾

Another example of applied mathematics is to model asset prices as continuous quantities, while knowing that real prices have a finite number of decimal places, given e.g. in cents.

Epilogue

In summary, mathematics is the intellectual discovery of nature's infrastructure. It consists of theories about ideas that we sense with varying vagueness.

A theory begins with a number of ideas and assumptions, from which its story unfolds via the derivation of more and more logical consequences. The entire process happens within one's mind, relying fully on one's intuition.

The aim in mathematics is to reflect deep laws of nature; that's where its beauty comes from, and that's how it is connected with arts.

What is still missing is a clarification of the terms "logical consequence" and "fundamental idea", which were both used informally all along. Discussing these in detail will be the topic of another essay.

Related reading

M. Balaguer, E.N. Zalta [ed.] (2011), Fictionalism in the Philosophy of Mathematics
E. Bishop, D. Bridges (1985), Constructive Analysis
J. Bolyai (1831), F. Kárteszi [ed.] (1987), Appendix: The Theory of Space
D. van Dalen [ed.] (1981), Brouwer's Cambridge Lectures on Intuitionism
K. Devlin (1993), The Joy of Sets: Fundamentals of Contemporary Set Theory
H. Field (2008), Saving Truth from Paradox
H. Field (1980), Science Without Numbers: A Defence of Nominalism
T. Gowers [ed.] et al. (2008), The Princeton Companion to Mathematics
J.L. Heiberg [ed.] (1883-1885), R. Fitzpatrick [trans.] (2008), Euclid's Elements of Geometry
D. Hilbert (1926), E. Putnam & G.J. Massey [trans.] (1964), On the infinite
D. Hilbert (1898-1899), E.J. Townsend [trans.] (1902), The Foundations of Geometry
D.R. Hofstadter (2008), I Am a Strange Loop
H.J. Keisler (2011), Foundations of Infinitesimal Calculus
E. Nelson (2005), Completed versus Incomplete Infinity in Arithmetic
E. Nelson (2010), Confessions of an Apostate Mathematician
E. Nelson (1977), Internal Set Theory: A new approach to nonstandard analysis
E. Nelson (2000), Mathematics and Faith
E. Nelson (1987), Radically Elementary Probability Theory
E. Nelson (2002), Syntax and Semantics
E. Nelson (2006), Warning Signs of a Possible Collapse of Contemporary Mathematics
B. Russel (1919), Introduction to Mathematical Philosophy
E. Schechter (2001), Constructivism Is Difficult
I. Stewart (2010), Alien mathematics: is Pi universal?
P. Suppes (2001), Finitism in geometry
P. Suppes (2000), Quantifier-free axioms for constructive affine plane geometry
P. Suppes (2002), Representation and Invariance of Scientific Structures
P. Suppes (2010), The nature of probability
M. Tiles (1989), The Philosophy of Set Theory: An Historical Introduction to Cantor's Paradise
E.P. Wigner (1959), The Unreasonable Effectiveness of Mathematics in the Natural Sciences
F. Ye (2011), Strict Finitism and the Logic of Mathematical Applications

FAQ

(1) Isn't "threeness" crystal clear?
Good question. I don't think it is.

(2) Is √-1 a ubiquitous idea?
No, it's neither fundamental nor ubiquitous, but it bears a close relationship to such ideas, which at the end of the day makes it interesting for mathematical discovery. On a related note, the various kinds of mathematical spaces (vector spaces, topological spaces, etc.) are all about discovering what emerges from the interplay between the fundamental, ubiquitous ideas of set, relation, and operation.

(3) What about discoveries where computers are used?
There is a philosophical concept of the "extended mind", which for the mathematical practice means that the mind can be aided by things like pencil and paper, calculator, or computer. Such extensions boost already existing capabilities of the mind, especially the memory and the ability to derive logical consequences.

(4) What if it turns out that a story is not true?
No theory is true, there are only circumstances under which it cannot be falsified. The story will continue to have its own life in our mind and can be further developed via logical deductions.

(5) Wouldn't it be more productive to allow experiments in mathematics?
Thought experiments are allowed, of course. Others would be of very limited use, since they would either be rendered superfluous by logical proofs (due to the "unreasonable effectiveness" of logic in mathematics), or just couldn't be performed at all (e.g. due to infinity involved).

(6) What does "disproved" mean in constructivism?
A statement is disproved after a contradiction has been derived from the assumption that it's true (= proved). This is how the truth value "false" is defined. The following are synonymous: "A is false", "A is disproved", "not A". To paraphrase, a statement is false if and only if it has been proved that the situation described by it cannot occur.

(7) Assuming that the square root of 2 has not yet been constructed, can we say that it does not exist, or only that its existence is undecided?
If existence is meant literally, then given the assumption, the statement "the square root of 2 exists" is false. However, if by "exists" we actually meant "can be constructed", then the statement is (currently) undecided.

(8) Why are constructive proofs cumbersome?
I believe it has to do with our education, namely that we have been trained in classical mathematics from an age of 6 or so. As a result, textbooks on classical mathematics can be written in an informal style where many details that would clutter the main line of thought are omitted. It does not compromise rigor, since it capitalizes on the readers' solid understanding of and intuition about the fundamentals. Alternative mathematical theories don't have this luxury. In their textbooks, theorems and proofs are stated either in great detail at the expense of increased clutter, or with less details risking that the audience gets confused or misunderstands what is written. Either way, the exposition is likely to be difficult to follow.

(9) Does strict finitism deny the existence of infinity?
No, not necessarily. It rather says that we don't need to care whether infinity exists or not, because we can still do useful mathematics without it. As of today, the resulting theories seem sufficient for the purpose of modeling the finite world we know about.

(10) Is "applied mathematics" mathematics at all?
Yes, because in the models we work with mathematical ideas. However, the application of a model is unlikely to yield deep mathematical discoveries, for the aim is to solve problems of another discipline, not that of mathematics.