steppi (u/steppi) - Readit News

steppi commented on Approaching 50 Years of String Theory math.columbia.edu/~woit/w... · Posted by u/jjgreen

gnfargbl · 18 hours ago

People often say that the problem with string theory is that it doesn't make any prediction, but that's not quite right: the problem is that it can make almost any prediction you want it to make. It is really less of a "theory" in its own right and more of a mathematical framework for constructing theories.

One day some unusual observation will come along from somewhere, and that will be the loose end that allows someone to start pulling at the whole ball of yarn. Will this happen in our lifetimes? Unlikely, I think.

steppi · 10 hours ago

I was planning to make a similar comment. Conjecturing that some theory in the string theory landscape [0] gives a theory of quantum gravity consistent with experiments that are possible but beyond what humans may ever be capable of isn't as strong of a claim as it may first appear. The intuition I used to have was that string theory is making ridiculously specific claims about things that may remain always unobservable to humans. But the idea is not that experiments of unimaginable scale and complexity might reveal that the universe is made up of strings or something, it's just that it may turn out that string theory makes up such a rich and flexible family of theories that it could be tuned to the observed physics of some unimaginably advanced civilization. My impression is that string theory is not so flexible that its uninteresting though. There's some interesting theoretical work along these lines around exploring the swampland [1].

[0] https://en.wikipedia.org/wiki/String_theory_landscape

[1] https://en.wikipedia.org/wiki/Swampland_(physics)

steppi commented on Mathematicians don't care about foundations (2022) matteocapucci.wordpress.c... · Posted by u/scrivanodev

black_knight · 12 hours ago

Yes, but most mathematicians do not seem to make this distinction between sturdy and flimsy truths. Which puzzles me. Are they unaware? If so, would they care if educated? Or do they fully commit to classical logic and the axiom of choice if pushed? I can see it go either way, depending on the psychology of the individual mathematician.

steppi · 12 hours ago

I don't think they usually make the distinction in a formal sense, but I think most are aware. The space of explorable mathematics is vastly larger than what the community of mathematicians is capable of collectively thinking about, so a lot of aesthetic judgment goes into deciding what is and what isn't interesting to work on. Mathematicians differ in their tastes too. A sense of sturdiness vs flimsiness is something that might inform this aesthetic judgment, but isn't really something most mathematicians would make part of the mathematics. Often, ones interest isn't the result itself, but some proof technique that brings some sense of insight and understanding, and exploring that often doesn't make much contact with foundational matters.

steppi commented on Mathematicians don't care about foundations (2022) matteocapucci.wordpress.c... · Posted by u/scrivanodev

black_knight · 13 hours ago

This seems to me to be the same as saying that mathematicians do not care about the meaning of their theorems. That they are only playing a game. They care about consistency only because inconsistency means one can cheat in their game.

I know TFA says that the purpose of foundations is to find a happy home (frame) for the mathematicians intuition. But choosing foundation has real implications on the mathematics. You can have a foundation where every total function on the real numbers is continuous. Or one where Banach–Tarski is just false. So, unless they are just playing a game, the mathematicians should care!

steppi · 12 hours ago

I'd say that I care deeply about the meaning behind theorems, but just find results which swing widely based on foundational quirks to be less interesting from an aesthetic standpoint. I see the most interesting structures as the ones that are preserved across different reasonable foundations. This is speaking as someone who was trained as a pure mathematician, moved on to other things, but tries to keep up with pure math as a hobby.

steppi commented on There is No Quintic Formula [video] youtube.com/watch?v=9HIy5... · Posted by u/DamnInteresting

CamperBob2 · 20 days ago

Looks like a nice book, but what's up with his assertion on page 148 (164 of the .pdf) that the integers don't form a group under addition?

If he defines integers as "natural numbers excluding zero," that seems goofy and nonstandard but also interesting. Is that a Russian-specific convention?

steppi · 20 days ago

It seems like a typo where "integers" is used when the intention was to write "natural numbers". That is the solution to exercise 194 part a) which asked if the set of natural numbers is a field.

steppi commented on There is No Quintic Formula [video] youtube.com/watch?v=9HIy5... · Posted by u/DamnInteresting

steppi · 21 days ago

Vladimir Arnold famously taught a proof of the insolubility of the Quintic to Moscow Highschool students in the 1960s using a concrete, low-prerequisite approach. His lectures were turned into a book Abel’s Theorem in Problems and Solutions by V.B. Alekseev which is available online here: https://webhomes.maths.ed.ac.uk/~v1ranick/papers/abel.pdf. He doesn't consider Galois theory in full generality, but instead gives a more concrete topological/geometric treatment. For anyone who wants to get a good grip on the insolubility of the quintic, but feels overwhelmed by the abstraction of modern algebra, I think this would be a good place to start.

steppi commented on Calculus for Mathematicians, Computer Scientists, and Physicists [pdf] mathcs.holycross.edu/~ahw... · Posted by u/o4c

godelski · a month ago

It certainly is more painful, but it is more beneficial. It is also harder to teach, but I stand by my claim.

I'll quote Poincare:

  Math is not about the study of numbers, but the relationships between them.

The difficulty and benefit of the rigor is the abstraction. Math is all about abstraction.

The abstraction makes it harder to understand how to apply these rules, but if one breaks through this barrier one is able to apply the rules far more broadly.

  ----

Let's take the Fundamental Theorem of Calculus as an example[0]:

  f'(x) = lim_{h->0} {f(x + h) - f(x)} / {h}

Take a moment here and think about it's form. Are there equivalent ones? What do each of these symbols mean?

If you actually study this, you may realize that there are an infinite number of equations that allow us to describe a secant line. So why this one? Is there something special? (hint: yes)

Let's call that the "forward derivative". Do you notice that through the secant line explanation that the "backward derivative" also works? That is

  f'(x) = lim_{h->0} {f(x) - f(x - h)} / {h}

You may also find the symmetric derivative too!

  f'(x) = lim_{h->0} {f(x + h) - f(x - h)} / {2h}

In fact, you see these in computational programs all the time! The symmetric derivative even has the added advantage of error converging at an O(n^2) rate instead of O(n)! Yet, are these the same? (hint: no)

Or tell me about the general case of

  f'(x) = lim_{h->0} {f(x + ah) - f(x + bh)}/{(a-b)h}

I'm betting that most classes that went through deriving the derivative did not answer these questions for you (or you don't remember). Yet, had you, you would have instantly known how to do numerical differentiation and understand the limits, pitfalls, and other subjects like FEM (Finite-Element Methods) or Computational Methods would be much easier for those who take them.

  ----

Yet, I still will say that this is much harder to teach. Math is about abstraction, and abstraction is simply not that easy. But abstraction is incredibly powerful, as I hope every programmer can intuitively understand. After all, all we do is deal with abstractions. One can definitely be overly abstract and it will make a program uninterpretable for most, but one also can make a program have too little abstraction, which in that case we end up writing a million variations of the same thing, taking far more lines to write/read, and making the program too complex. There is a balance, but I'd argue that if one is able to understand abstraction that it is far easier to reduce abstraction than it is to abstract.

This is just a tiny taste of what rigor holds. You are absolutely right to be frustrated and annoyed, but I hope you understand your conclusion is wrong. Unless you're Ramanujan, every mathematician has spent hours banging their head against a literal or metaphorical wall (or both!). The frustration and pain is quite real! But it is absolutely worth it.

[0] Linking an EpsilonDelta video that covers this exact example in more detail https://www.youtube.com/watch?v=oIhdrMh3UJw

steppi · a month ago

Just a couple of corrections.

Let's take the Fundamental Theorem of Calculus as an example[0]:

  f'(x) = lim_{h->0} {f(x + h) - f(x)} / {h}

This isn't the Fundamental Theorem of Calculus, it's the usual definition of the derivative of a function of a single real variable. The Fundamental Theorem of Calculus establishes the inverse relationship between differentiation and integration [0].

Unless you're Ramanujan, every mathematician has spent hours banging their head against a literal or metaphorical wall (or both!)

Ramanujan was no stranger to banging his head against the wall. My impression from Kanigel's The Man Who Knew Infinity is that his work ethic and mathematical fortitude were as astonishing as his creativity. For much of his career, he couldn't afford paper in quantity and did his hard work on stone slate, only recording the results. This could make it seem like his results were a product of pure inspiration because he left no trace of the furious activity and struggle that was involved.

From The Man Who Knew Infinity:

When he thought hard, his face scrunched up, his eyes narrowed into a squint. When he figured something out, he sometimes seemed to talk to himself, smile, shake his head with pleasure. When he made a mistake, too impatient to lay down his slate pencil, he twisted his forearm toward his body in a single fluid motion and used his elbow, now aimed at the slate, as an eraser. Ramanujan's was no cool, steady Intelligence, solemnly applied to the problem at hand; he was all energy, animation, force.

[0] https://en.wikipedia.org/wiki/Fundamental_theorem_of_calculu...

steppi commented on Dead soldiers' teeth reveal diseases that doomed Napoleon's army washingtonpost.com/scienc... · Posted by u/reaperducer

readthenotes1 · 2 months ago

"one that would see his army decimated by cold, hunger and disease."

2 things: A)decimated means 1 in 10, not 9 in 10. B) according to the wiki article, Napoleon had already lost 75% of his initial fighting force by the time he got to Moscow, before the withdrawal.

I am not sure an article on biology should include much history--I would certainly hope it did a better job on the biology...

steppi · 2 months ago

From https://www.merriam-webster.com/dictionary/decimate

Decimate is a word that often raises hackles, at least those belonging to a small but committed group of logophiles who feel that it is commonly misused. The issue that they have with the decline and fall of the word decimate is that once upon a time in ancient Rome it had a very singular meaning: “to select by lot and kill every tenth man of a military unit.” However, many words in English descended from Latin have changed and/or expanded their meanings in their travels. For example, we no longer think of sinister as meaning “on the left side,” and delicious can describe things both tasty and delightful. Was the “to kill every tenth man” meaning the original use of decimate in English? Yes, but not by much. It took only a few decades for decimate to acquire its broader, familiar meaning of “to severely damage or destroy,” which has been employed steadily since the 17th century.

steppi commented on A recent chess controversy chicagobooth.edu/review/d... · Posted by u/indigodaddy

indigodaddy · 3 months ago

General statistical question. If we say extend the coin flip example distribution to say 10B times. Should/would we expect to see a streak of 100 or even 1000 in the distribution somewhere? Intuition alone tells me probably not for 1000 but a smallish chance for 100 (even if 10B in a row i would think a streak of 100 would be unlikely)

steppi · 3 months ago

Your intuition's not bad. The expected value for the longest run of heads in N total flips of a fair coin is around log2(N) - 1 with a standard deviation that's approximately 1.873 plus a term that vanishes as N grows large. log2(10B) - 1 is approximately 32 and with that standard deviation, even a run of 100 in 10B flips is incredibly unlikely. For more info see Mark F. Schilling's paper, "The Longest Run of Heads" available here https://www.csun.edu/~hcmth031/tlroh.pdf.

steppi commented on Bit is all we need: binary normalized neural networks arxiv.org/abs/2509.07025... · Posted by u/PaulHoule

bjourne · 3 months ago

Yeah, but then there is no performance benefit over plain old sgd.

steppi · 3 months ago

Yeah, I think surrogate gradients are usually used to train spiking neural nets where the binary nature is considered an end in itself, for reasons of biological plausibility or something. Not for any performance benefits. It's not an area I really know that much about though.