A month ago, the loop quantum gravitist Carlo Rovelli wrote a somewhat surprisingly intelligent essay on Platonism in mathematics,

Michelangelo's Stone: an Argument against Platonism in MathematicsI consider myself a Platonist but I do agree with about 1/2 of Rovelli's observations. Some of them are "sort of obviously true" while others are even "innovative and persuasive". The remaining 1/2 is composed of propositions that are either "non sequitur" or "widespread misconceptions" or "downright ludicrous". ;-)

First, just to be sure, Platonism is the philosophy believing that the world of (interesting/important/natural/beautiful) mathematics exists independently of us and we are only "discovering" it, not "inventing" it or "creating" it.

Top mathematicians typically tend to consider themselves Platonists. Rovelli quotes Penrose and Connes but he could mention many other, less media-savvy mathematicians. But his interesting example of a de facto Platonist is Michelangelo who argued that he wasn't creating statues. He just picked the statues from inside the stone where they had already pre-existed.

OK, Rovelli's first three major points are that

- we have to ask how inclusive the "empire of mathematical structures and results" is, and if it is too inclusive, it will be mostly a dumping ground
- consequently, we have to care which of the structures are "picked"
- once we realize the dependence on our "choice", the nature of the "empire of mathematics" inevitably depends on our identity – which subsequently contradicts or disproves the Platonist independent existence of mathematics

Obviously, some selection is needed because "most" of the axiomatic systems and theorems in them that we could write down are uninteresting, irrelevant, or contrived in one way or another.

OK, the rest of Rovelli's paper is that he looks at examples of mathematical structures and insights that should "obviously" be a part of the "empire" and he offers argumentation that the incorporation of those things depended on our or someone's random history, environment, and other things. He claims that those observations – that the culture or practical needs accelerated or decelerated some discoveries or the interest of the people in these discoveries – disproves Platonism.

Well, I think that this conclusion of Rovelli's is silly. Every Platonist knows that the real world – and especially the human society – is messy. The history is full of coincidences, non-holy pressures, and no human being – and even no top mathematician – is quite perfect so that he would know what is "out there" in the idealized empire of mathematics and how important it's there.

But this is not what Platonism wants to claim, anyway. Instead, the very purpose of Platonism is that these questions have some "idealized" objective answers that are independent of us and the progress in mathematics – and the opinions of the deepest mathematicians – are just converging towards that ideal. I don't think that Rovelli has actually disproved

*this*true interpretation of Platonism. He may have only disproved his straw man.

Even

*that*result is questionable, however.

When he starts to discuss the examples, he wants to assign labels "Yes" and "No" to topics and questions in mathematics to indicate which of them make it to the idealized "empire of important mathematics". I don't think that this "Yes/No" attitude is a very sensible one. Instead, different insights or axiomatic sets or formulae or identity or algebraic structures or classifications or other theorems should be assigned some "real numbers" that indicate the relative "power" – you may literally view it as the percentage of their power – within the "empire of mathematics".

So some game theory has some "power" within that realm. But if you start to ask increasingly technical features of some increasingly contrived games – generalizations of chess or something like that – it's very clear that the relative power of these insights in the world of "deep mathematics" decreases. It decreases sufficiently quickly for "all things related to chess" to maintain only a tiny portion of the power in the "realm of mathematics".

Rovelli continues by an argument that the Euclidean geometry is linked to some specific economic needs in Egypt when it was inhabited by the ancient Greeks. This linkage of the Euclidean geometry to Egypt is just totally ludicrous. Every civilization above a certain threshold has to master the Euclidean geometry because the latter is an important limit that describes tons of things. It holds for civilizations on Earth but it would hold for extraterrestrial civilizations, too.

Babylonia and Vedic India had their own industry around the "Euclidean" geometry. The ways how they talked about all these matters obviously weren't isomorphic to the Greek ones but they knew something. And if they have ever gotten to the level of the Greco-Roman Antiquity, which they arguably have not before the worlds began to interact again, they would have to be familiar with all the theorems, too.

Rovelli's claim about the "provincial" character of the Euclidean geometry sounds so ludicrous to me because I believe that this portion of mathematics would be essential not only in our Universe but all other Universes that remotely resemble ours – certainly in all compactifications of string/M-theory that admit intelligent life.

His following argument is more interesting. He argues that the spherical geometry, especially the triangles on a sphere, are more natural and beautiful than those on the plane. I actually found his arguments convincing. But that doesn't disprove the broader claim about Platonism. If the spherical geometry actually has a greater power in the "Platonism empire of mathematics" than the planar geometry, it's how it is, and every human who thinks otherwise may be just making a mistake.

The coolest difference showing how the spherical geometry is simpler concerns the formulae for the areas of triangles. Note that triangles on the sphere are composed of three "maximally straight" geodesics on the sphere (arcs whose curvature radius agrees with the radius of the sphere \(R\)).

The area of a planar triangle with sides \(a,b,c\) is\[

A = \frac{ \sqrt{ 2a^2b^2+2b^2c^2+2c^2a^2-a^4-b^4-c^4 } }{4}

\] This is a pretty messy formula. For spherical triangles, we can write the area in this way:\[

A = R^2 (\alpha+\beta+\gamma- \pi).

\] Isn't it absolutely cool? Note that for infinitesimally small triangles, the spherical geometry reduces to the planar one where \(\alpha+\beta+\gamma=\pi\) so the expression would vanish. But the larger the triangles are, the greater the deviation of the sum of angles from \(\pi\) becomes. The deviations grow with the second powers of the lengths in such a way that this extremely simple formula works. An elegant method to prove the formula is to divide a larger triangle into infinitesimal ones and realize that the "excess angles" from the small shapes are adding up.

(Also, check the formula for the "octant". The angles are \(\alpha=\beta=\gamma=\pi/2\) and indeed, you get the area \(4\pi R^2 / 8 = \pi R^2/2\).)

The simplification of the formula for the area reminds me of the simplification of the Poisson bracket in quantum mechanics. The formula for the Poisson bracket is somewhat messy but in quantum mechanics, it's superseded by the simpler commutator \([A,B]=AB-BA\). Isn't it similar to the simplification of the formula for the area of triangles?

Too bad, triangles themselves don't seem to be the "core of geometry" as understood by physicists. Triangles are just some simple yet arbitrary subsets of the geometric spaces and physics isn't so much about "subsets". Only when matter becomes a continuum, like condensed matter, we may talk about "subsets of space" (objects). But in physics, we believe that they're not too fundamental. (Needless to say, when I was 7, I did believe in a "theory of everything" which was a classical theory remembering the exact shape of some bodies.)

Again, the particular point is great. But it may convince others that the spherical geometry is deeper than the planar one – in the same sense in which quantum mechanics is deeper than classical physics. And if it is so, it may change our understanding how it works in the Platonic empire of mathematics, too!

Rovelli also claims that linear algebra – the basis of quantum mechanics – is something extremely simple and important that great civilizations should appreciate right away, just like Heisenberg did. Can there be a civilization that didn't appreciate the complex Hilbert spaces right away, he asks? Yes, our civilization is an example.

Again, I agree that the discovery and appreciation was delayed too much and some people still fail to appreciate the linear algebra. But I don't think that this fact weakens Platonism itself. Heisenberg's and pals' discoveries simply moved us closer from the messy world of deluded mammals to the idealized world of mathematics.

Finally, Rovelli thinks about a hypothetical civilization composed of fluids on Jupiter which has no solid creatures. He claims that they won't know integers because everything is continuous.

I don't share his intuition at all. A set of physical objects that won't learn about integers won't be able to do anything else that is intelligent, according to my understanding of "intelligence", either. Some "discrete mathematics" may fail to be fundamental but it's ultimately needed to organize the data and even the knowledge about the objects, both discrete and continuous ones. So if the things on Jupiter were as fluid as Lee Smolin, they would never surpass his intelligence, either, and I think it's correct to approximate this intelligence by zero.

A point that Rovelli misses is that even when the "fundamental structures" are continuous, the integers and discrete structures inevitably emerge. After all, the fundamental physical laws as we know them may be defined in terms of objects that are a priori continuous – like fields in quantum field theory. But they inevitably have lots of "discrete" implications. Also, continuous functions such as the Riemann zeta function – whose importance may be justified by purely continuous considerations – inevitably have lots of connections to primes and number theory in general.

Even in the Jovian world of Rovelli's, integers unavoidably arise. If there is a civilization where everyone is made of liquids, well, one can still count droplets – the number of components of a manifold. And even if the boundaries of such objects were fuzzy, we may count the components in the approximation in which the fuzzy objects resemble the sharp ones. We can count the number of holes and many other topological invariants. If there's any interesting "life" out there, the creatures will have to do this counting, anyway.

So at the end, I think that Rovelli's examples pretending to show that these important parts of mathematics are only analogous to the "God-Creator who just resembles our grandfather sitting on the cloud" are deeply flawed. We may be imperfect in our knowledge of the Platonist realm of mathematics and some "friends of ours" whom we consider powerful entities in the empire may actually be trumped by some of them that we don't know or we do know them and underestimate them (spherical geometry) but all this imperfection may be considered to be our fault, not the evidence against Platonism as a philosophy.

The comparison with the "grandfather on the cloud" is surely funny. But this analogy is also a demagogic one because the evolution of people's opinions about "the important structures in mathematics" was getting less and less anthropomorphic, just like physics was getting less anthropomorphic. So even if I agree that the "world of deep mathematical ideas" should look (almost) nothing like the humans, and I do agree with that, Rovelli actually hasn't presented any evidence against Platonism because the evolution of the shapes of the realm of mathematics according to the Platonists have been going in the opposite direction than Rovelli suggests – away from the naive anthropomorphic opinions and towards the more abstract and universal ones. And it's the asymptotic future than the Platonist considers closer to the truth.

I have to run. This text may be proofread tonight. (Update, I did it on Monday morning.)

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