peterhalburt33

Wish I had time for a longer comment, but this advice rings very true for me. Reflecting back on grad school and the transition to professional life, you have to realize that your role changes every couple of years and that the things that got you to one stage won’t get you to the next. Many people end up stuck in a local maximum (lacking vision) which partly explains the Peter principle.

As a mathematician who’s only recently started to get into computation and programming, I think the difference between my thought patterns when switching hats is so fascinating.

I was so accustomed to hearing that mathematics is nothing if not rigorous, but the more I reflect, mathematics is much more dependent on social convention and agreement amongst a community. While an outsider might think that proofs rigorously establish theorems, the purpose of a proof might be better seen as having enough detail to convince a substantial portion of the prominent mathematicians in a field that the proof is correct. In fact, there are theorems (e.g. the ABC conjecture) where a “proof” has been proposed, but not enough mathematicians have expertise with the techniques used to prove it in order to agree whether the proof is sufficient or not (though I’ve heard that the general opinion is that the proof does not suffice). William Thurston wrote one of my favorite essays related to this topic: https://www.math.toronto.edu/mccann/199/thurston.pdf

Reflecting on my own experience in mathematics, a better way to think of proofs is as being composed of “thought patterns” which many mathematicians agree are likely to be correct - when I scan a proof, I don’t look through every detail to verify that it is correct, but rather run it through a series of high level tests to see if it fails in any way, then if it passes all of those I look more closely at the argument and analyze the structure and mathematical power of each statement (e.g. one is unlikely to establish a hard analytic result through purely algebraic means, so where is the magic going on?) and so on until I’ve convinced myself that the argument is probably true. Other times, the result may be “visually apparent” (e.g. in geometry) at which point it might be sufficient for me to just to connect certain canonical arguments with the pictures as I read through the proof. For an excellent overview of this process, read Terry Tao’s blog on identifying errors in proofs : https://terrytao.wordpress.com/advice-on-writing-papers/on-l....

I don’t feel as confident commenting on the programming/computational perspective, as I’ve probably developed a very idiosyncratic way of thinking from approaching the topic so late in my education, but my feeling is that they are much different, and that the types of things a mathematician wants to convey to another mathematician rely much more on “trust” rather than the kind of rigor that might be needed by a computer.

I think this would be an interesting topic to explore in longer form.

I was just about to say, I was slightly disappointed to find that this wasn’t an article about a novel technique to identify fast moving water (akin to the fact that we can identify the temperature water by the sound it makes when it is poured).

Not Math books, but David Griffiths Electrodynamics and Quantum Mechanics are such a pleasure to read for their style of writing.

Wait, what? The last thing we want is more Elizabeth Holmes types…

As a mathematician who works with many engineers and computer scientists, I wanted to expand on one of the points under the “Getting a Job” section. While it is certainly true that a mathematics education provides a great background for understanding other STEM fields, I would caution math Ph.D students who expect these jobs to be open to them because of their STEM connection: the onus is completely on you to bridge the gap between what you do and the field you want to work in. While it may be true that someone will hire you for your critical thinking skills (though I will personally say that I have never seen this happen), it is more likely that your deep specialization in a tangentially connected field (coupled with not being involved in the culture/conferences of the community you wish to enter) will be an impediment to entering a new field: you expect to be paid like a Ph.D., but will potentially require years of training to get up to speed.

As an example, I remember the advice of “just go into data science” being handed out like candy to students interested in industry around the time I was in grad school (10 years ago). To be sure, there was a period where a STEM background + interest could get you in the door, but that time is over. These days you are competing with many equally brilliant students who have taken multiple courses and done research in this area, and it is highly unlikely that an employer will take a chance retraining an e.g. algebraic geometer with no precious data science experience to suit their needs.

All this to say, if you have an interest in another area, you must know the players and their work in that area while simultaneously knowing your area in math. It is not easy by any means, you are essentially signing up for twice as much work learning your field and theirs, but the rewards are great - as a connector between two fields, you have precious expertise that is very employable across a broad range of industries (my first job out of grad school was essentially providing advice on research programs helping connect different STEM communities to government funding agencies, but I was able to use my connections from that job to get back into research).

It aggravates me too, however, considering that the definition of a tensor according to mathematicians is an element of the tensor product of two vector spaces (or whatever other objects you can tensor together), and according to physicists would be an object which transforms like a tensor, I’m somewhat sympathetic. Neither definition sheds any light on what a tensor is to anyone who doesn’t already understand what a tensor is, and I’m convinced that the moment one understands what tensors are they lose the ability to explain what tensors are.

I will say IMO (and experience) in professional math that while there is perhaps more of a chance for an outsider to have an impact, Mathematics is hardly free from bias towards insiders: it can manifest itself as subtly as using notation as a shibboleth (e.g. it’s somewhat easy to tell which community an author comes from through their notation and terminology, and equally easy to harbor resentment towards those outside your field) all the way to active “prove I’m the most clever in the room” syndrome during seminars. I’d like to think that a more collaborative atmosphere is prevailing now due to the rise of interdisciplinary and applied math, but people are people everywhere and as Sayre stated “Academic politics is the most vicious and bitter form of politics, because the stakes are so low.”

I would also mention David Graeber’s excellent “Bullshit Jobs”

To your point, how can one measure 1% improvement in a meaningful way for abstract activities (e.g., learning)? The premise is that 1% improvement each day will yield large gains through the power of compounding, but if you aren’t sure you are making a quantified 1% improvement over your previous improved state things go awry quickly: if you only make 1% improvement over your original state each day then by the end of the year you have only improved by a factor of 4.65x, rather than your projected 37.8x improvement. Point being, if you want to take advantage of the sensitivity and power of exponential growth, you better be sure that you have a good way of quantifying your growth.

I know this is probably meant to be more inspirational than quantifiable, but then why even insert numbers unless to mislead people about the amount of effort it takes to improve? Either way you have to put in the same amount of effort to achieve the improvement, whether you break it up into 1% chunks or not there’s no free lunch.

The first time I attempted Rudin I gave up after the second chapter. I had been through Abbot, and still didn’t truly understand Analysis (or the style of analysts). After letting it sit for a few months, I picked Rudin back up and it instantly became my favorite text. I have no clue why this change happened so suddenly, but as a convert I will point out that he is a master at including exactly the right amount of detail: he does not leave details out, but neither does he commit the equally (if not more) harmful sin of over explaining a concept. This leaves every concept explained as concisely and simply as possible, but no more, and is extremely helpful as a person with ADHD. The book is worth it’s weight in gold: one can read a chapter of Rudin in a day and have a better understanding of the concept than reading twice or three times as much material in a competing analysis text.

I’d agree, but most first year students aren’t ready to work at the level of abstraction required for a rigorous LA course (i.e., more than just matrices and Gaussian Elimination etc.)

The equations you quoted result from minimizing the square of the norm of the residual of Ax-b over all inputs x, so in a sense least squares is just calculus…

I’d push back slightly on the idea that a PhD isn’t an optimal route to industry careers.

Due to the variety of experiences and knowledge I acquired in grad school, I became conversant in a few different fields, which allowed me to cross disciplines to a very nicely compensated career path WITHOUT meeting many of the relevant criteria that new grads might be subject to.

So sure, maybe a PhD isn’t optimal if one values money and is certain that they are on a lucrative career path, however, it has opened doors for me to enter a career path where individual enrichment is a large focus since the prestige of the company is built upon the accomplishments of individual employees.

If nothing else, I got to spend five years enriching myself, studying what I love and getting paid (modestly) to do it. While there are certain aspects of academia that are off putting, grad students are mostly shielded from politics and get to spend time focused on learning their field and becoming experts with no serious expectation of a useful deliverable at the end. You might argue that the thesis is the deliverable, however, since one is working at the boundary of knowledge, there’s often very little oversight and one can justify chasing down exploratory research directions that turn out to be dead ends. My experience in industry is that it is more a zero sum game, where you have to carve out opportunities to enrich yourself and hours and productivity are tracked much more closely.

Agreed this isn’t airtight, but if you accept the premise that real numbers are specified by their sequence decimal digits: .499…+.499… has 9 as its first digit after the decimal, 9 as its second and so on - the only other number that has this decimal expansion is .999… infinitely repeating, so these must be the same numbers.

You can compute it digit by digit:

.999…/2 = .45 + .045 + .0045 + …

Completely correct. Too lazy to amend my answer, but even rational numbers don’t have a unique decimal representation.

I know this might be a somewhat long and tangential answer, but I wanted to give a different viewpoint, because this is such an unintuitive fact.

Firstly, let me just say, for how much time we spend learning about the number line and calculating with Real numbers, they aren't the cute and familiar number system you think they are. Beneath the facade of familiarity lie a whole bunch of technical constructs and counterintuitive facts which justify the suffering of many undergraduates taking their first course in Real analysis (and the existence of books such as "Counterexamples in Analysis").

Lets start off by trying to define the real number system so that we can agree what a real number actually is - it's not unreasonable to think that .999... might be a fundamentally different object than 1, perhaps belonging to a set of "approximate" numbers. Without discussing any of the technicalities, I think a reasonable first stab at a definition of the Reals between could be the set of all decimal numbers (e.g. 111... or 7.500...).

On to my main point, how should we define equality of two real numbers? The most naively appealing answer would be through equality of their decimal representations (i.e. two numbers are equal if and only if their decimal expansions are equal, and if the decimal representations are different then the real numbers are different). Under this viewpoint, each real number has a unique decimal expansion (since real numbers are in one to one correspondence with decimals, and different expansions mean different numbers), and .999... is not equal to 1 (or 1.000...) because their decimal representations are different.

However, since there are no "gaps" in the Real number line, there must be some other number between the two. What would the decimal representation of this number look like? If .999... and 1 are truly different, then the mean should lie strictly between the two.

(1 + .999...)/2 = .5 + .499... = .999..

Well that's frustrating. What about trying to "squeeze" another number between the two decimal digit by decimal digit? The first decimal digit of this number has to be 9 (since it is less than 1, but greater than .999...), and by the same reasoning so must the second digit, and on and on...

Perhaps 1 - .999... is an "infinitesimal" real number given by .000... infinitely repeating followed by a 1 at the end. If you believe this, let's try and write out the decimal expansion of this number digit by digit. Of course the first decimal digit is 0, along with the second, and the third and so on. From this information, the decimal expansion of this "infinitesimal" number is a string of 0's (i.e. 0). You might object that I'm not considering the 1 at the end: what you really meant was a sequence of numbers getting smaller and smaller i.e. whatever number the sequence .01, .001, .0001 and so on tends to. But if this sequence is to represent a real number, it must have a single unique decimal expansion at the end of the day, and there is no escaping that all of the digits must be 0.

At this point, it should start to be apparent that the idea that every real number has a decimal expansion, and that the expansion must be unique (i.e. Real numbers with different decimal expansions are different numbers) are in conflict with each other. But why should we define equality through representation? After all, 1/3 and .333... infinitely repeating define the same number, but have quite different representations.

I hate to be reductive, but for whatever reason mathematicians have decided that the benefits of allowing the real number system outweigh the drawbacks of allowing non-unique representations of these numbers. In fact, one sees this idea repeated over and over throughout mathematics e.g. completeness of Lp spaces outweighs the drawback of generalizing measure and defining Lp functions as equivalence classes equal a.e., expanding the definition of derivatives to allow for weak differentiability permits a wider class of solutions to PDEs (such as shocks) and so on. There's always some sort of trade-off to be made between nice behavior and power + generality, and while it is often painful to adapt to, it pays off in dividends to go beyond one's intuition in mathematics.

I think Intel would rather we all forget about netburst.

It really depends on which field you are talking about. I’d say it is very hard to find an area of math that’s completely new , but you will often find existing areas where novel perspectives are driving math forward. Geometric algebra may be a hot topic in some areas, but the ideas of exterior algebra go back more than a century at this point, so is it really new??

Just to humor you though, I think Deep Operator learning is a vastly exciting new field which combines ideas from functional analysis and deep learning in order to do things like solving PDEs.

I love Terry Tao’s writing on math. One thing that strikes me about him is that, despite being an absolute technical powerhouse, he writes in a very down to earth style that connects disparate areas of math - e.g. his article on “what is a gauge” https://terrytao.wordpress.com/2008/09/27/what-is-a-gauge/am... where he explains how dimensional analysis might be viewed as a change of coordinates. Too often exposition in math is myopic and fails to impart a unique perspective on the subject, but Tao imbues his writing with a wisdom that I consider the sign of a true genius.

Exactly.

As someone who has been privileged to be naturally inclined towards the more lucrative parts of STEM, I still don’t want to see a society which rewards people like me in a vastly outsized way, while being indifferent or punitive to others whose interests or abilities or life experiences landed them outside a handful of in-demand professions. It’s very easy to attribute success to the choices we make, but so much of it is out of our control (where we live, what our society values, what advantages and disadvantages we inherit).

Yeah, it’s funny how most of high school is spent focusing on the “exceptional” solvable cases of nonlinear equations (low degree polynomials, simple trig equations, exponentials) that one can come out with the skewed idea that solvability in nonlinear equations is more common than it actually is. While I understand that it’s important to build up a vocabulary of basic functions (along with confidence manipulating them) I think it is also important to temper expectations with the reality that nonlinear behaviors are so diverse and common that it is an a small miracle that we have somehow discovered enough examples of analytically solvable systems to enable us to understand a rich subset of behaviors!

One of my favorite perspectives on the difficulty of formulating a general theory of PDE in light of the difficulties posed by nonlinearities is Sergiu Klainerman’s “PDE as a unified subject” https://web.math.princeton.edu/~seri/homepage/papers/telaviv.... If I understand correctly, any general theory of PDE would have to incorporate all the subtle behaviors of nonlinear equations such as turbulence (which has thus far evaded a unified description). Indeed, ”solvable” nonlinear systems in physics are so special Wikipedia has a list of them https://en.m.wikipedia.org/wiki/Integrable_system. With this perspective, I’m tempted to say (in a non-precise manner) that solvable systems are the vanishingly small exception to the rule in a frighteningly deep sea of unsolvable equations.

Huh, somehow I also learned LA best through Griffiths QM. I almost wish Griffiths would write a Linear Algebra textbook.

It really depends on what you’d like to learn LA for, and how comfortable you are with abstraction: LA can span from concrete multiplication of matrices and vectors all the way to very abstract (e.g. vector spaces over general fields or even modules over rings). I know many people recommend Gilbert Strang’s introductory linear algebra (I have not read it, but it seems to fall into the former camp), but I might also recommend Sheldon Axlers Linear Algebra Done Right. In all honesty, I learned La the best from David Griffiths quantum mechanics text, although it is not a comprehensive in its coverage of the subject (not that it should be, given that it is a physics text). I guess I am trying to say that there are many different flavors and interpretations of linear algebra, and while matrices and vectors may be simple at first, it does tend to rob the subject of its richness and depth (e.g. what do these matrices represent, what are the canonical structures etc.) so I am a bit biased towards going full generality at first, and perhaps reading a more rote computation book on the side (I understand we all have limited time though).

I would also add that one fundamental aspect of linear algebra (that no one ever taught me in a class) is that non-linear problems are almost never analytically solvable (e.g. e^x= y is easily solved through logarithms, but even solving xe^x=y requires Lambert’s W function iirc). Almost all interesting real world problems are non-linear to some extent, therefore, linear algebra is really the only tool we have to make progress on many difficult problems (e.g. through linear approximation and then applying techniques of linear algebra to solve the linear problem).

Teaching only statistics and no calculus is how you end up with people such as Tai reinventing high-school calculus and attempting to use statistical methods to validate their “method”.

https://math.berkeley.edu/~ehallman/math1B/TaisMethod.pdf

They will always be the Unreal Tournament company to me. Unreal and Return to Na Pali are still two of my favorite games to this day. Honestly wish someone would make games like those again.

I’d be hard pressed to recommend D&F to a self study audience. Perhaps it’s just me, but the endless enumeration of examples and counterexamples at the beginnings of most chapters seemed to destroy my intuition rather than reinforce it.