[calhoun137]

> I expect that mathematics will be rewritten to suit computing, rather than vice versa

I agree with this. I believe pure mathematics is suffering greatly because many mathematicians refuse to fully embrace the computational power of modern technology.

My belief is the age of pretty formulas is coming to an end, and that the future of mathematics will be it focuses more and more on computational aspects of the subject, and problem sets in pure math courses will be done using programs that are much more advanced than anything which exists today, and everyone will think nothing more of those programs than we do about calculators.

Apologies for the self plug, but this has been my vision with mathinspector[1]. I've been working very hard on that, and this is why I got so interested in your statement. Thank you for clarifying your thinking here. Makes sense to me, and you could be right

[1] https://github.com/MathInspector/MathInspector

[randomNumber7]

> I agree with this. I believe pure mathematics is suffering greatly because many mathematicians refuse to fully embrace the computational power of modern technology.

They can't even give meaningful names to their variables. When showing code to mathematicians we should always rename all variables with only one character. Then look down at them because they can't understand it ;)

[calhoun137]

I think the reason mathematicians use single letter variables is because mathematics is about the way that people think about patterns, understand patterns, and the relationship between them. Therefore the letter is used to help our brain understand the connection between abstract concepts (Categories).

As opposed to programming languages, where the goal is to do something practical, in pure mathematics, our goal is to create a language capable of helping our brains understand the infinite complexities of nature.

https://math.stackexchange.com/questions/24241/why-do-mathem...

[GregarianChild]

The MathInspector is nice. Reminds me of the "Incredible Proof Machine" [1] which I find to be a neat tool in teaching logic.

[1] https://incredible.pm/

[calhoun137]

Thank you!!! It's so funny that you mentioned "scriptable notebooks like Jupyter or Mathematica" since I have been spending all of my time on mathinspector recently.

I think our points of view are actually very strongly aligned. However I believe the next big idea is likely to come from outside of computer science.

Personally, I am betting on biology. So many of the most sophisticated techniques are based on biology, e.g. neural nets and genetic algorithms. I have done a lot of work on extending the theory of computation with a new axiom which gives Turing machines a self replicating axiom[1], [2]

In many parts of science, there is a cross pollination, where new ways of thinking about subject X come from a new discovery in subject Y. Typically, research will follow a group think pattern until it hits a brick wall, then you need that really big breakthrough idea. This line of reasoning leads to the conclusion, imo, that it's approximately equally likely to come from either pure computer science, or pure mathematics, or somewhere else.

[1] https://math.stackexchange.com/questions/3605352/what-is-the...

[2] https://medium.com/swlh/self-replicating-computer-programs-8...

[GregarianChild]

It's hard to predict the future.

Alan Turing invented neural nets in a little-known paper entitled Intelligent Machinery (see [1]), in 1948. Since, the use of NNs has moved away decisively from inspiration by nature. I reckon, nature's last big win in AI were convolutional NNs: Kunihiko Fukushima's neocognitron was published in 1980, and inspired by 1950s work of Hubel and Wiesel [2]. Modern deep learning is largely an exercise in distributed systems: how can you feed stacks and stacks of tensor-cores and TPUs with floating point numbers, while minimising data movement (the real bottleneck of all computing)?

Not unlike, I think, how airplanes were originally inspired by birds, but nowadays the two have mostly parted ways, for solid technical reasons.

[1] http://www.alanturing.net/turing_archive/pages/Reference%20A...

[2] https://en.wikipedia.org/wiki/Neocognitron

[calhoun137]

> take the average Joe so much beyond what even Gauss, Newtown, Grothendieck or Archimedes had at their disposal

I think this comment really sums up very well what is at the core of our discussion: the future of mathematics and science.

My strong belief is that thousands of years from now, Archimedes and Gauss will still be remembered, and everything we think is great now will be forgotten while they are not. That tells me that they were much farther ahead of their times than us, even though they didn't have modern computers.

Mathematicians and computer scientists both have it totally backwards imo. On the one hand, mathematicians think they have something to teach us about computer science, but they refuse to use technology properly. On the other hand, when we write code, it's all governed by mathematical laws and there are many questions (but maybe not you know, coding standards or the philosophy of writing good code) we could really use the guiding hand of mathematicians with, and they need to catch up with the times and we programmers need to accept they have something valuable to offer and to teach us.