tprice7

THat's not one of the recommendations he gives. That's not even part of the email itself.

Which one of the recommendations given here do you think would entail workaholism?

A completely different basis free definition, that works whenever you are working over the complex numbers or any other algebraically-closed field:

The determinant is uniquely determined by the following two axioms:

1. The determinant of the "multiply by lambda" operation on a one-dimensional vector space is lambda. 2. If you have a linear operator T on a vector space V, and a T-invariant subspace W, the determinant of T is the product of the determinant of T restricted to W and the determinant of the operator that T induces on the quotient space V / W.

This actually kinda intuitively meshes with the volume-stretching property: if you are stretching the volume by the factor lambda_1 along one subspace and by the factor lambda_2 along some complementary subspace, clearly the overall stretch factor is lambda_1 * lambda_2.

If you aren't working over an algebraically closed field, you can just tensor with the algebraic closure of whatever field you're working with and take the determinant there. There is also a way of adapting the definition so you don't have to do this, but it makes axiom (1) a bit more complicated.

Another bonus is this definition makes the Cayley-Hamilton theorem completely trivial.

Also, you can give an analogous definition of the trace if you replace multiplication with addition, and of the characteristic polynomial if you replace lambda in axiom 1 with x - lambda.

"Usually" So there is an instance where this isn't the case?

But don't these models all have a true zero, distinct from the infinitesimals, which is its own negative?

Do you have a reference for this? I would believe you if you said "an infinitesimally small value" rather than 0. If it's true then I'm very surprised. For one, 0 = -0, so if sin(0)/0 = 1, then -1 = -sin(0)/0 = sin(0)/-0 = sin(0)/0 = 1

The two situations are not analogous, there is a crucial difference.

Like tekromancr was getting at, you get a logical contradiction if you assume the existence of 1/0, if you also assume the ring axioms. To elaborate, if 0x = 1, then 0 = 1. It is also an axiom that 0 != 1, and even if you abandon this axiom you only get one extra ring that only has one element. Boring!

You also get a logical contradiction if you add a square root of -1, if you are assuming, say, the ordered ring axioms. But you can lose the order structure and then there is no more contradiction.

If you want to add a new number x with 0x = 1, then you also need to propose how one might alter the notion of a ring so that you don't get a logical contradiction, and you are still left with something interesting. Just like how we altered the notion of an ordered ring, by removing the ordering, so that you don't get a contradiction when adding a square root of -1.

So the crucial difference is, as far as I know, nobody has succeeded in doing this when it comes to division by zero.

"Renate Scheidler, a math professor at the University of Calgary, said there’d be no six-string guitars, so music would sound different"

Good thing they asked exclusively math professors for a response, so you know you are getting an authoritative answer from someone with many years of training and deep knowledge about the topic. LOL

"Even if you're using very little of what you think of as "math" to interact with your machine learning framework, to dismiss math as not being necessary is ignoring all of the underlying mathematics that went into creating it."

This seems like a straw man to me. The article never claimed that math is not necessary for the development of machine learning frameworks. In fact, it explicitly states the opposite:

"First of all, math is particularly important if you’re doing machine learning research in an academic setting."

"In particular, there are people at companies like Google and Facebook who are pushing the boundaries of machine learning – people working on bleeding edge tools. These people almost certainly employ calculus, linear algebra, and more advanced math routinely in their work."

Ok, so you are saying that, since people in general like to do fun things, it takes a lot of self discipline to make yourself learn programming, history, etc, which in general requires doing things that aren't fun. I think you are overlooking something however: people don't just like to do fun things, they also like to make things that are cool (maybe not all people but this is very common in my estimation). And making things that are cool often requires doing boring work and learning boring things, even though the end result is not boring. I find this can be a very powerful source of motivation, so it actually might not take so much discipline as you think.

I would even speculate that how interesting / boring you find something is not static, but that otherwise boring things can start to seem interesting once it's clear that they are relevant to something you want to achieve.

Personally I learnt math mostly from self-study, and I've learnt it well enough that I now have published research (see [1], also [2] is in review and has been recommended for acceptance with revisions by the referee), have been invited to give talks about my research [3][4][5], and have stayed for a few months as a visiting researcher at University of Muenster [6]. I don't consider myself as having a lot of self-discipline; I have a tendency to hit the snooze button many times, to overeat, to let my room get messy, etc. But once I have an idea, I want to flesh it out and share it with other people, which motivates me to do the things that are necessary for this, which are sometimes boring.

[1] http://content.algebraicgeometry.nl/2017-2/2017-2-007.pdf

[2] https://arxiv.org/abs/1612.07306

[3] https://www.uni-muenster.de/FB10srvi/HotNews/show_artikel.ph...

[4] http://www2.math.binghamton.edu/p/seminars/arit/arit_spring2...

[5] http://www2.math.binghamton.edu/p/seminars/arit/arit_fall201...

[6] https://web.archive.org/web/20170121000748/https://wwwmath.u...

In relation to Bill Sethare's work, I recommend listening to example (1) here : http://sethares.engr.wisc.edu/html/soundexamples.html , which is an amazing demonstration of the relationship between overtone beating and consonance / dissonance.

Even if the statement "lack of beating between overtones sounds good" doesn't hold true as a universal rule (and I'm sure this has occurred to many people prior to Wilkerson), you can really see what Helmholtz was getting at here.

Hmm interesting. So it seems the Boltzmann machine explanation of dreams that I described becomes less convincing when you consider that running the waking and dreaming phases of the training algorithm in separate long "chunks", instead of simultaneously, seems to not work well in practice.

"I wouldn't be so sure that Boltzmann machines were designed without how the brain operated in mind."

Sure, actually it seems highly unlikely to me that they were designed without considering at all how they brain works. Let me clarify what I meant: given the definition of how a Boltzmann machine infers (i.e. how to determine which neurons are on or off), the training algorithm can be derived from purely mathematical considerations.

As someone with no neuroscience background I’ve suspected this for a while just based on how Boltzmann machines work. These are a type of unsupervised artificial neural network.

The training algorithm for Boltzmann machines has two phases:

1. (Awake phase) Feed the network training data, and increase the weights between neurons that tend to get activated simultaneously in this phase.

2. (Dreaming phase) Let the network run without any training input, and decrease the weights between neurons that tend to get activated simultaneously in this phase.

This training algorithm wasn’t designed to emulate how human brains work, it just sort of falls out of the math. And yet (1) resembles the Hebbian principle from neuroscience (neurons that fire together, wire together), and (2) resembles the process described in this article. Actually I suspect something stronger than what the article claims based on this: that the connections that get pruned are the ones connecting neurons that fire together while dreaming. It would be nice to have someone who knows stuff about neuroscience comment on this.

You made an error: p1p2...pn + 1 is not necessarily prime, but none of its prime factors are in {p1, ..., pn}. For example, 15 = 2 * 7 + 1 is not prime, but neither 3 nor 5 are in {2, 7}. I agree with your overall point though.

I don't get anything when I ctrl-F for "moment" or "density" there (other than in sidebar links).

I am curious which particular proofs you are referring to, could you please give a link or reference or something?

"there isn't a truly compelling reason to believe that RH is true, aside from numerical evidence and a belief in beauty."

You wouldn't consider the analogue of the Riemann hypothesis for the zeta function of a smooth projective variety over a finite field, proven by Deligne, to be a compelling reason to believe the RH is true?

Is there some objective sense in which conservation of momentum is more fundamental than conservation of energy in a different reference frame? Since either can be derived from the other, how do you determine which is "actually a consequence" of which? (I ask out of genuine curiosity)

Did you see my response to the parent, where I give a sketch of how conservation of momentum follows directly from conservation of energy in all frames of reference? What is your objection to that argument?

I had never realized that conservation of momentum is just a special case of conservation of energy, thank you for sharing this interesting observation! (I see now that, since the change in energy when changing the frame of reference is determined by the dot product of the momentum with the velocity of the new frame of reference, conservation of momentum follows from conservation of energy in _all_ frames of reference)

'The question for me is, does it deliver more, less or the same amount of knowledge than during Higgs's time?'

I don't think that is the right question. Say it produces more knowledge, but the fields that are studied are selected, not by how intrinsically interesting they are, but instead by how much low-hanging fruit they have. I don't think that would necessarily be an improvement.

If you don't consider the topics mungoid mentioned to be advanced math, perhaps you'll still consider this to be: http://arxiv.org/abs/1509.05797 It was accepted to Algebraic Geometry in May, (the Foundation Compositio Mathematica journal, not JAG), so it will probably appear online sometime around December. So far I've received three invitations to visit two different universities as a result of this paper. (search for "Price" on these pages: http://www2.math.binghamton.edu/p/seminars/arit http://www2.math.binghamton.edu/p/seminars/arit/spring2016 , I will also be travelling to a German university in October but unfortunately I have no evidence to show for this currently). I say this as someone who left undergrad after four terms and is mostly self-taught, from such resources as books, online papers, and wikipedia.

Yes, I've encountered one form of the Curry-Howard correspondence before. Are you claiming that, via the Curry-Howard correspondence or some related phenomenon, the corresponding proof of a program used to compress a proof is the same (or at least, proves the same theorem) as the original proof that the program was meant to compress? That is interesting and I would appreciate if you could expand on it.

Maybe we could define the Kolmogorov complexity of a proof as the number of lines of code, PLUS the size of the proof that the proof-generating program halts.

Or the total number of lines of code in a sequence of programs, where the first program generates a proof of the original theorem, and every other program generates a proof that the preceding program halts, PLUS the size of a proof that the final program halts.

"As the new shorter proof is the compressed representation + the decompression program." A proof is a sequence of deductions, not a program that generates a sequence of deductions.

That's an interesting idea and piece, thanks for sharing.

Based on rewqfdsa's comment history, this is probably sarcasm.

I think MattHeard understood this and his point was that the weakest man will realistically have quality C + epsilon + random, and the weakest woman will have quality C + K + epsilon + random. The random term arises because no evaluation process is going to perfectly tell you how people are going to end up performing. But yeah, this seems fixable also, by averaging some number of the lowest performing members of each group, as MattHeard suggested.

Another inaccuracy: the sound example "sine, saw, triangle, square" only correctly gives an example of a sine wave. The other three sounds are neither saws, triangles, or squares, but these weird irregular waveforms, which sound completely different. Open it up in Audacity and see for yourself.