What is it that is done when we DO mathematics?

Question

I want to understand more deeply and philosophically what exactly mathematicians do.

Wikipedia lists some major subareas like analysis, geometry but ends its lead paragraph with

There is no general consensus among mathematicians about a common definition for their academic discipline.

Clearly at the physical level, mathematicians make sounds with their vocal cords and draw figures with chalk and type on keyboards. We of course understand that that cannot constitute mathematics since other activities like writing an English essay may have the same elements.

So then what are the deeper contents of the doing of mathematics?

Answer 1

1. What are the deeper contents of the doing of mathematics?

   Mathematicians invent concepts from the domains below, state some interesting axioms and prove by applying logical rules, that certain statements about these concepts are true.

   It is not far away from a game with rules made up by the mathematicians themselves.

2. Domains of mathematics: An influential group of mathematicians under the pseudonym Bourbaki set out in the midst of the 20th century for a new approach in mathematics The Architecture of Mathematics.

   Bourbaki did not care about labels like Platonist, formalist, intuitionist etc.. The group build mathematics around the concept of structure. Bourbaki identified three basic structures in the field of mathematics

   • Algebraic structure
   • Order structure
   • Topological structure

   Around these structures Bourbaki published a series of books, covering many important domains of pure mathematics: Theory of Sets, Algebra, General Topology, Functions of a real variable, Topological vector spaces, Integration, Commutative Algebra, Lie Groups and Lie Algebras, Spectral Theory. Bourbaki’s enterprise has not been finished.

   Books by Grothendieck and his collaborators like EGA (Elements of Algebraic Geometry) and SGA (Seminar of Algebraic Geometry) add further material in new fields. Also these volumes remain unfinished.

   The basic idea of Bourbaki is to ask for the structure, which mathematicians incorporate into the objects they define, and to study those maps, which respect these structures. E.g. any map f between vector spaces should be a linear map, i.e. satisfying

   f(x+y) = f(x)+f(y) and f(alpha * x) = alpha * f(x) , alpha a scalar while x,y vectors.

   The idea of a map between objects, which respects the structure of the objects, has been generalized under the name “morphism” within a new mathematical discipline named Category theory.

Added: I changed the order of the two sections.

Answer 2

Short Version

To me the central thing about math is that it is platonism-reified.

When a mathematician does mathematics what they are doing is performing that reification.

Note the fine dance that's already in evidence here:

• The mathematICS lives in the platonic realm
• Doing the math is in the empirical realm
• The mathematiCIAN is a dancer between the two

Longer version

While we are here primarily talking of platonism¹ in math we need to start at least a bit with Platonism by itself ie. Plato qua Plato.

Platonism

A very brief graphical outline of Platonism is this.

Its broad contours may be described thus:

• We inhabit two worlds
  • called by Plato — the sense-ible and the intelligible worlds
  • called after Plato — the physical and the Platonic
  • in more modern terminology — the Empirical and the Rational
• The physical is what we directly see/perceive but is just shadow, not real. The intelligible that we don't see but is the real can — if we try and are rightly educated — be intuited
• Philosophy is that practice (notice the empirical hiding here?) of sharpening those intuitions.

This understanding is so simple yet overwhelming in its scope that Whitehead said of it:

All of western philosophy is just footnotes to Plato.

It remains 2 millennia later in Kant's phenomenon-noumenon and is sufficiently universal to be found cross culturally, vide Vedanta's Maya-Brahman.

But what does this have to do with math?? you may ask!

Right — Lets hear it from Plato himself. This is the inscription over his academy:

Let none but mathematicians enter here²
Plato

Now this (should) rightly confuse one who's a bit familiar with the history: Plato was a philosopher! Why does he insist on math?

Because just as speaking English, dressing decently, having enough means to commute, buy books etc is a prerequisite to enrolling in a school, the same way according to Plato, math-ability is a prerequisite to philosophical ability.

Structure of Platonic Realm

If one looks up mathematical platonism one would find discussions on how/whether math-entities like '2' and '+' reside in a so-called eternal Platonic heaven where irrespective of the contents of the mind of any mathematician, 2+2 = 4. and this is so eternally.

But Plato hardly talks of math, instead he talks of things like the realm where the eidos³ of truth, beauty, justice etc and ultimately the eidos of the Good eternally reside.

So this intelligible-Platonic world has an interesting structure: Its inhabitants are the eidos of beauty justice etc. Its bottom element, most accessible to us stuck in the empiric shadow world, is the eidos of Truth and the highest — effectively God for Plato — is the eidos of the Good.

Process of Platonic philosophising

What Plato taught can be outlined into three steps:

1. Plato ultimately wishes us the eidos of the Good. This is the goal but is ultimately outside the reach of philosophy
2. To apperceive that, we must first reach the eidos of Truth — the essential contents of philosophy
3. To reach even there we need to think true thoughts, for which math, actually doing math, is a fine preparation.
   [Plato adds astronomy and music — see quote at end]

A feeling for this eternality of math can be perceived in the well known:

Mathematics, rightly viewed, possesses not only truth, but supreme beauty a beauty cold and austere, like that of sculpture, without appeal to any part of our weaker nature, without the gorgeous trappings of painting or music, yet sublimely pure, and capable of a stern perfection such as only the greatest art can show. The true spirit of delight, the exaltation, the sense of being more than Man, which is the touchstone of the highest excellence, is to be found in mathematics as surely as in poetry.

Bertrand Russell

So clearly you understand that scribbling and making sounds may be associated with math but is itself not math. You need to take that further: While the layperson may see a mathematician calculating, computing, proving etc, what the mathematician is really doing behind the appearances is delving in the platonic (or even Platonic) realm.

As an analogy consider how as children, we learnt to count using our fingers. As adults we — hopefully! — dont consider fingers necessary or germane to math. So in the same way, not just are the greek sounds and strange diagrams incidental but not necessary, even the more fundamental (seeming) calculating, computing, proving are only incidental, the core being the apperception of the platonic realm.

Yet all this may sound strange and far away from us because of...

The Inversions of Modernity

Modern entrées into math are almost always as a part of STEM — as the last component to boot. This implies that it has been relegated from Queen of science to Handmaiden of Science.

At the end of Hardy's active mathematical life he wrote the famous Mathematician's apology in which he said:

I have never done anything "useful". No discovery of mine has made, or is likely to make, directly or indirectly, for good or ill, the least difference to the amenity of the world.

Now its true that the pugnacious tone is because Hardy was a pacifist and he lived across two world wars. Yet, if we look beyond the exaggerated wistful, poetic tone there is an important germ of truth — insofar as math is 'good', it must disengage from the world and approximate the Platonic realm. If it engages too much, it may make for effective engineering but it's likely to be pedestrian math.

And speaking of Hardy its apposite to refer to his most famous protege:

An extremal Platonist: Ramanujan

Ramanujan is one of the most extreme examples of a Platonist-mathematician. Ramanujan outright saw the Goddess dictate number theory results in bright red vermilion. And its not relevant to my point if, say, we find out that Ramanujan had never heard of Plato. That he had a clear sense of a realm from which his results flowed makes him a Platonist.

People may object that if we take a hundred mathematicians hardly ten would have even the most basic knowledge of Plato. So how can one claim that all mathematicians are platonists?

Well the meaning of platonism has very little to do with the word 'platonism'. Even the philosophy of Plato is at most incidental.

As long as the mathematician believes that there is something called mathematical truth they are a platonist.

ie The mathematician may play symbol games but he is not merely a formalist in saying its all just symbol games.
Nor is he merely an intuitionist who understands math as a peculiar property of the human mind. Sure math is reflected in the mathematician's brain but the reflection is the shadow while the truth is 'out there'.

And even the most prosaic of mathematicians who believe in math being discovered — even if the psychological fact of doing math comes closer to invention — are platonists.

The alternative is this — funny in movies; frightening as it veers into 'our' reality.

So let me end with one more Plato anecdote:

Xenocrates wanted to study with Plato without knowing music or geometry or astronomy
Plato: Go, because you do not have the handholds of philosophy

---

¹ Following a suggestion by Conifold, I use "platonist" in the general sense of anyone who believes in pre-existing (mathematical) truth. And "Platonist" in the stronger sense of Plato's own commitments to a spiritual realm from where truth flows. Even amongst professional philosophers such punctiliousness is not systematically practiced. But it's good to at least try to keep the distinction given that I expect many more people here will belong to the larger category than the smaller!

² I've taken slight liberties here. The Greek original is ἀγεωμέτρητος μὴ εἰσίτω. "Only geometers may enter." Given that math was effectively geometry in Greek times and is considerably wider today, I've taken his "geometers" to mean our modern "mathematician"

³ The currently fashionable English word for Plato's greek eidos is 'form' but I wont use that since it's a really horrible translation of something which is closer to idea → ideal → essence → soul

Answer 3

One answer is provided by Leibniz in terms of his idea of mathematical entities as "useful fictions". The view of mathematical entities as fictional was propounded by mature Leibniz no later than 1676 (in his early period one finds all sorts of alternative views, not grounded in a deep knowledge of mathematics; see this publication). This goes hand-in-hand with a Formalist attitude that has common elements with that of Hilbert's Formalism, as argued in several publications by the philosopher David Sherry and myself; see here.

When Galileo said that the book of the universe is written in math symbols (which was a novelty for his time), or Gauss said that math is the queen of all the sciences, presumably what they meant was that mathematical tools are developed to solve problems of the natural sciences. Today this would still be the viewpoint of applied mathematicians (though with a broader interpretation of "natural sciences"). But how would a pure mathematician describe the entities he is studying?

One possibility would be to take a Platonist stance (as in this answer). Another possibility would be to use the analogy with art. Historically, art was representational. Abstract art of the modern variety has dispensed with the requirement of representing anything; yet artists today identifiably work in ways that are similar to earlier generations in many aspects.

Answer 4

As a first stab at an answer, based on the operations of my own mind, I would argue that if the mind is not recognizing the sense of proportion and/or the concept of numbers then it is not doing math. The rules for manipulating symbols are not math if the results are incoherent with the sense of proportion and the concept of numbers on a basic or advanced level of comprehension.