Thread by @davidbessis

My most ambitious post ever: we've been *wrong* about math for 2300 years and I’m going to *fix this* in a single Twitter thread.

From the time of Euclid, there has been two competing approaches to defining math:
1/ through what it studies: numbers, shapes, structures…
2/ through how it functions: axioms, theorems, logical deduction…

These definitions reflect the two prevailing philosophies of mathematics:
1/ Platonism: mathematical objects “exist” in the ethereal realm of ideas.
2/ Formalism: mathematics is a mechanical game of syntactic deduction with zero transcendent semantics.

Both philosophies are dead-ends, as expressed by Reuben Hersh brilliant quip that “the working mathematician is a Platonist on weekdays and a formalist on Sundays.”
(Quote is from his beautiful 1979 paper:
) sciencedirect.com/science/articl…

Like many mathematicians, I had this weird feeling that something was deeply broken in the traditional way of presenting math.
I thought there was nothing I could do about it, because the issue seemed ancient and unfixable, and it almost felt like a structural feature of math.

This frustration is a key driver of our desire to engage the general public and attempt to redefine the public perception of math. This is why we insist on using words (such as “poetry”, “joy”, or “love”) that aim to reset the expectations (ping @stevenstrogatz).

@stevenstrogatz But like Hersh, I’m of the opinion that the issue isn’t just a matter of public perception: we need to **fix** the definition of math, and the only way to do so is to get rid of deeply ingrained yet untenable philosophical traditions.

@stevenstrogatz I feel that we are at an interesting moment in the history of math where we're actually going to find a valid definition:
1/ because it needs to be done,
2/ because it may finally become possible.

@stevenstrogatz About 1, let me simply say this:
Just like you can’t market a product if you can’t explain it, you can’t teach a subject if you can’t define it.
If we can’t define math, it’s no surprise we’re having so much trouble teaching it.
In the end it’s **our** responsibility.

@stevenstrogatz Why am I saying it might be possible to fix it now? Because we’re reaching a time where the legacy philosophical nonsense about thought, cognition and language is finally dissolving.
In the age of ChatGPT and Zoloft, unexamined dualism can only remain fashionable for so long.

@stevenstrogatz Mathematicians can’t yet agree on a definition of math, but they already share a latent consensus about what it means to do math and what it feels like.
“Mathematica, a Secret World of Intuition and Curiosity” is my attempt to document this consensus. amazon.com/Mathematica-Se…

@stevenstrogatz I entirely agree with Reuben Hersh’s deep conclusion that math can only be defined as “certain kind of human mental activity”. This also very well aligns with Bill Thurston’s take that “mathematics is that which mathematicians study”.
arxiv.org/pdf/math/94042…

@stevenstrogatz However, I believe we must go one step deeper and detail the specifics of math as a “certain kind of human mental activity”.
This is what I’m trying to do in my book.

@stevenstrogatz Personally, I wasn’t able to make sense of math while I was stuck in the Platonism vs Formalism debate and its “Quarrel of Universals” counterpart: the Realism vs Nominalism debate.
en.wikipedia.org/wiki/Problem_o…

@stevenstrogatz My book was heavily influenced by my personal encounter with Deep Learning and my feeling that it offers an empirical way of resolving the Quarrel of Universals: Abelard’s Conceptualism (a flavor of Nominalism) has won.
en.wikipedia.org/wiki/Peter_Abe…

@stevenstrogatz What does it mean in practice? You can’t define math if you don’t recognize that cognition is a dynamic learning process.
Math is a particular way of orchestrating the interplay between language and intuition.

@stevenstrogatz When we do math, we play the “Truth Game”: we make *as if* words had a stable meaning, statements had an absolute truth value and we could infer new valid statements from existing ones by pure logical deduction, without any empirical evidence.

@stevenstrogatz This weird and seemingly crazy way of playing Legos with definitions is unique to math and provides its foundational Formalist flavor.

@stevenstrogatz But if you’re a Conceptualist, you don’t believe that words have a transcendent meaning, and it seems even more bonkers that playing the Truth Game is of any use.

@stevenstrogatz This is where the neural learning aspect kicks in: as it happens, when we play this game, our intuitive understanding of the underlying concepts actually solidifies into something extremely powerful.

@stevenstrogatz My hunch is that the “unreasonable effectiveness of math” follows from a machine-learning theorem: the Truth Game is an effective way of accelerating the convergence of a deep learning network.

@stevenstrogatz In the end, mathematicians are Platonists on weekdays because math works. It makes their intuitions so crystal clear that they get the feeling that they can **touch** the abstractions they’re manipulating.

@stevenstrogatz Successful math is math that no longer feels like math. Nobody feels that whole numbers are abstractions, yet they are.

@stevenstrogatz In Chapter 4 of my book, I use the example of “a billion minus one”: everybody can see the result in their head. But 2000 years ago nobody could see it, because it’s incredibly hard to write with Roman numerals.

@stevenstrogatz That’s all folks, thanks for bearing with me!