Everyone is capable of, and can benefit from, mathematical thinking

Some of us are satisfied with, say, dimensional reductions of spectral geophysical surveys to improve imaging and train that against existing production records to highlight potential mineral leases.

Others might like to leverage symbolic algebra systems to crack quantum encryption candidates.

A good number like to get rich via high frequency trading, some like to sail satellutes against the magnetic currents of the planet, there's a world of optimisations in logistics that have been implemented and are still being fine tuned to improve throughput efficiency and fuel consumption.

But you're not really asking in good faith here, are you?

I know this reply may not suffice to convince you, but unfortunately I won't be able to argue forever.

Did you ever consider the possibility that you might be the one living in a bubble?

FYI, the concept of innate talent predated IQ tests and twin studies by many millenia. Two of the authors I'm citing in my book (Descartes and Grothendieck) believed that innate talent existed and they both declared they would have loved to be naturally gifted like these or these people they knew.

You're declaring that these incredibly smart people were wrong about their own domains, which is a pretty bold claim to make. What do you have in support of this claim? A fake Einstein quote?

It's a sad fact of life that most quotes attributed to Einstein are fabricated. Next time, please check "The Ultimate Quotable Einstein", compiled by Alice Calaprice.

This may come as a shock to you, but Google page 1 isn't always a reliable resource. Nor is Wikipedia, even though it's quite often correct. As it happens, there's a pretty large "Heritability of IQ" bubble on the internet. It's active and vocal, but it's also quite weak scientifically — the page you're citing is a typical symptom, and it absolutely doesn't reflect the current scientific knowledge.

The IQ heritability claims that you're citing are based on twin studies and they have taken in serious beating in the past decade, especially in light of GWAS.

It's true that a number of people have been fooled by twin studies, most notably Steven Pinker, in Chapter 19 of the Blank Slate (did you read it?)

You see, Pinker is a linguist and apparently he isn't mathematically equipped to fully comprehend the intrinsic limitations of Bouchard's approach. Did you read Bouchard's 1990 paper on twins reared apart? Do you find it convincing? Are you aware that even The Bell Curve's Charles Murray thinks that this approach, abundantly cited by Pinker, is structurally flawed? Are you aware of the fundamental instability of IQ estimates based on twins reared together? Aren't you concerned that even a mild violation of Equal Environment Assumption, plugged into Falconer's equation, would drastically reduce the estimates?

If you don't understand what I'm talking about, if you've never read the authors and the primary research I'm citing, then it's quite likely that you're the one living in a social media bubble.

If you're interesting in learning more about the actual science of IQ heritability, I recommend using Sasha Gusev's Substack as an entry point: https://theinfinitesimal.substack.com/p/comments-on-no-intel...

Feel free to also subscribe to my own Substack, where I plan to cover these topics in the coming months: https://davidbessis.substack.com

All the best, David.

The book also agrees vehemently that math is NOT restricted to "geniuses" and even argues that those don't really exist in the way that culture thinks they do.

However! His assertion is that the (to him) tedious, laborious, error-prone, paperwork is not the fundamental output of "doing math". For him, symbolic written mathematics is akin to sheet music. It would in principle be possible to teach students to read and write sheet music and even do manipulations like transposing it to different keys, without ever letting them listen to music. It would be hard and boring. Some students would find the memorization and application of rules satisfying but most would struggle.

In such a classroom, there might be one student who by chance figures out for herself that you can kind of "hear" these symbols in your brain and suddenly all the arbitrary rules seem obvious and natural and she doesn't even have to go through the tedious steps at all to answer questions. "Of course this is in a minor key." she might say. "No, I didn't rigorously check each chord, it's just... obvious".

Such a student would be labeled a "prodigy" or "genius", and would struggle to explain to others that no, what she's doing isn't harder than the her classmates laboriously doing the rote work, it's actually much easier.

Of course... this is not to denigrate sheet music. It's a wonderful invention that makes it possible to transmit music out of one person's brain to the brains of an orchestra.

Just like written mathematics.

The author's contention is that, like the contrived example above, no-one ever talks about "the music" of mathematics, just the sheet music, and therefore things are much harder than they need to be.

One of the simple mathematical examples he uses is to ask: Can you imagine a circle in your head (unironically an amazing thing to be able to do!). Then to ask a question like: Can a straight line intersect a circle in 3 places?

You likely have an immediate, intuitive response to this highly non-trivial mathematical problem. That's the music. Now, try to write that down in mathematical language for someone who can't see circles. Oof, it's going to be a slog.

Much of this is just talking to oneself and testing it to see if our idea holds under test conditions.

I was once watching a video on how chess grandmasters think and work. Most of it is-

1. Do we know a pattern of moves, even if done, in series that is known to score some win/check. If so, lets do it.

2. Are any pieces under attack, If gone can effect point 1. eventually? If yes, lets protect them.

3. What can all possible moves of our pieces prevent opponent from having successfully execute their own point 1. And can we force opponent into point 2? Lets do it.

Basically every our move and its possible outcomes(Known through prior study of patterns of previous games seen), every move of our opponent.

A strong internal monologue and testing imaginary moves.

Math is just this except over paper.

For me, a progress bar or a pizza is the default. And because I cook rice daily and we were talking volume, my "progress bar" was like a measuring cup. Mentally, I can stand two measuring cups next one another, and filling one of two parts is less than filling two of three parts.

Maybe there are ways to be more aware of or insert ways to force the use of fractions in physical space. More cooking, more building, etc. The more comfortable you get the more intuition you should build