> The experts are in agreement that the important ideas are robust enough to withstand knocks like this, but the details of what is actually going on might not actually be where you expect them to be.

Past researcher in pure math here. The big problem is that mathematicians are notorious for not providing self-contained proofs of anything because there is no incentive to do so and authors sometimes even seem proud to "skip the details". What actually ends up happening is that if you want a rigorous proof that can be followed theoretically by every logical step, you actually need an expert to fill in a bunch of gaps that simply can't easily be found in the literature. It's only when such a person writes a book explaining everything that it might be possible, and sometimes not even then.

The truth is, a lot of modern math is on shaky ground when it comes to stuff written down.

> They may also recognize that eating humans comes with unknown risks because they are pretty observant of what humans are capable of.

Idk about that. We have some skills but orcas see basically none of them. From their perspective we are slow and bad swimmers who need to breath all the time.

Would an orca even understand that we made and control boats/ship as opposed to that humans live on boats/ships the same way fleas live on a dog?

Mind you, i’m not saying orcas are stupid. What I’m saying is that the slices of our life they can observe are not impressive, and the impressive things we do are not readily observable to them.

I'm the author of what you've just described as clickbait.

Interestingly, the 100m metaphor is extensively discussed in my book, where I explain why it should rather lead to the exact opposite of your conclusion.

The situation with math isn't that there's a bunch of people who run under 10s. It's more like the best people run in 1 nanosecond, while the majority of the population never gets to the finish line.

Highly-heritable polygenic traits like height follow a Gaussian distribution because this is what you get through linear expression of many random variations. There is no genetic pathway to Pareto-like distribution like what we see in math — they're always obtained through iterated stochastic draws where one capitalizes on past successes (Yule process).

When I claim everyone is capable of doing math, I'm not making a naive egalitarian claim.

As a pure mathematician who's been exposed to insane levels of math "genius" , I'm acutely aware of the breadth of the math talent gap. As explained in the interview, I don't think "normal people" can catch up with people like Grothendieck or Thurston, who started in early childhood. But I do think that the extreme talent of these "geniuses" is a testimonial to the gigantic margin of progression that lies in each of us.

In other words: you'll never run in a nanosecond, but you can become 1000x better at math than you thought was your limit.

There are actual techniques that career mathematicians know about. These techniques are hard to teach because they’re hard to communicate: it's all about adopting the right mental attitude, performing the right "unseen actions" in your head.

I know this sounds like clickbait, but it's not. My book is a serious attempt to document the secret "oral tradition" of top mathematicians, what they all know and discuss behind closed doors.

Feel free to dismiss my ideas with a shrug, but just be aware that they are fairly consensual among elite mathematicians.

A good number of Abel prize winners & Fields medallists have read my book and found it important and accurate. It's been blurbed by Steve Strogatz and Terry Tao.

In other words: the people who run the mathematical 100m in under a second don't think it's because of their genes. They may have a hard time putting words to it, but they all have a very clear memory of how they got there.

Everything you are looking for is provided in the Epilogue of Spivak. For instance, in chapter 28, you will be able to show that 0a = 0 in any field. In chapter 29, you will rigorously define the reals as well as the operations on them. You will also show that they form a complete ordered field, which allows you to use the results from chapter 28. In chapter 30 you will show that all complete ordered fields are "essentialy the same" as the real numbers---i.e., the real numbers are "unique."

At the moment I'd advise not worrying much about the construction of the reals. Ideas such as limits, continuity, differentiation, integration, and even fields are much more important for later mathematics and applications (abstract algebra, topology, geometry, physics) than the construction of the reals. Constructing the reals is pretty much something you do a couple of times (traditionally once with Dedkind cuts, as in Spivak, and once more with Cauchy sequences) and then never think about again.

Edit: I'm not sure what you mean by "something I could code." If you want something that you could type in a proof assistant you might have some luck looking at the mathlib library of Lean https://leanprover-community.github.io/mathlib4_docs/Mathlib....

> But the interviewers will not appreciate this solution approach lol.

I once witnessed a programmer with a PhD in Maths find closed form formulas for a lot of questions where it was expected to write some code with loops building/accumulating a result. As a simple example, to explain what was going on, if the question would be "calculate the 100th fibonacci number", she would just use Binet's formula to do so (as opposed to using a loop). I was rather impressed how often that happened.