Many aspects of Platonic thought can be traced back to the Pythagorean school. In particular, the almost religious views about mathematics, the deep mysticism, and the speculations about the workings of the world. Theatetus, as mentioned in the Platonic dialogues, was one of the best mathematicians of the time and was clearly among the Platonic school. His analysis of irrationals was fundamental for the time, and was one of the puzzles that the Pythagorean school had to address.
Agreed, Pythagoras - as a pre-Socratic precursor - had a very important formative influence on Plato's school:
The so-called Pythagoreans, who were the first to take up mathematics, not only advanced this subject, but saturated with it, they fancied that the principles of mathematics were the principles of all things.
—
Aristotle,
Metaphysics 1–5, c. 350 BC
With that being said, my personal understanding is that Pythagorianism is subtly (but quite majorly) different from the views of most mathematical Platonists in one critical respect.
Insofar as they are both mathematical mysticisms, Pythagoras appeared to teach - unless I've read him wrongly - that everything is
literallly mathematics: that physical reality was nothing but a huge mathematical totality - not just, as in Platonic thought, that everything physical can be described in the language of mathematics because mathematical forms / objects / relations 'pre-exist' independently as eternal abstract truths:
These thinkers [Pythagoreans] seem to consider that number is the principle both as matter for things and as constituting their attributes and permanent states.
Max Tegmark appears to be the modern theoretical physicist who is closest to pure Pythagorianism with his controversial and singular "
Mathematical Universe" hypothesis, which makes standard Platonic ideas look plausibly tame and modest by comparison:
Our Mathematical Universe by Max Tegmark – review
Close to his heart is an extreme Pythagorean/Platonic thesis: physical reality is ultimately nothing other than a giant mathematical totality.
So far, in parts one and two mathematics appears as no more than physics' indispensible tool kit for describing the world. The final part ups the ante: "Our reality isn't just described by mathematics – it is mathematics … Not just aspects of it, but all of it, including you." In other words, "our external physical reality is a mathematical structure". He calls this the Mathematical Universe Hypothesis (MUH).
Thus everything physical is ultimately mathematical, including you and me, "making us self-aware parts of a giant mathematical object".
Mathematical universe hypothesis - Wikipedia
Tegmark's MUH is: Our external physical reality is a mathematical structure.[3] That is, the physical universe is not merely described by mathematics, but is mathematics (specifically, a mathematical structure). Mathematical existence equals physical existence, and all structures that exist mathematically exist physically as well. Observers, including humans, are "self-aware substructures (SASs)". In any mathematical structure complex enough to contain such substructures, they "will subjectively perceive themselves as existing in a physically 'real' world".[4]
The theory can be considered a form of Pythagoreanism...a form of mathematical monism in that it denies that anything exists except mathematical objects; and a formal expression of ontic structural realism.
Tegmark claims that the hypothesis has no free parameters and is not observationally ruled out. Thus, he reasons, it is preferred over other theories-of-everything by Occam's Razor. Tegmark also considers augmenting the MUH with a second assumption, the computable universe hypothesis (CUH), which says that the mathematical structure that is our external physical reality is defined by computable functions.[5]
The MUH is related to Tegmark's categorization of four levels of the multiverse.[6] This categorization posits a nested hierarchy of increasing diversity, with worlds corresponding to different sets of initial conditions (level 1), physical constants (level 2), quantum branches (level 3), and altogether different equations or mathematical structures (level 4).
As both of us would agree, there are a lot of mathematical concepts which have no applicability to the primary world 'out there' but seem to just be pure mathematical - aesthetic conceptions of logic. Tegmark thinks, on the other hand, that all mathematical structures physically exist.
Tegmark's extreme Pythagoreanism is not what I personally believe and nor, in my estimation, would Plato have agreed with it (i.e. he was aware that only a 'subset' of the Eternal Forms - as he deemed these abstract objects / relations - applied to the physical world).
This leads Professor Tegmark to the (in my mind, at least) somewhat bizarre notion that
everything mathematical must exist out there 'somewhere' in the primary, physical world (in this string landscape / M-Theory 'multiverse'), whereas I think it's pretty clear that many maths concepts have no physical applicability in the natural sciences (and this fits with a Platonist framework, just as it does for different reasons in the Formalist one).
For justifiable reasons, I think pure Pythagorean philosophy is exceedingly rare today in maths / science / philosophy (unlike Platonism and Formalism, which remain widespread philosophical interpretations).
