peterhalburt33

3 yıl öncediscuss

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.

peterhalburt33

3 yıl öncediscuss

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.

peterhalburt33

4 yıl öncediscuss

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).

peterhalburt33

4 yıl öncediscuss

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

peterhalburt33

4 yıl öncediscuss

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

peterhalburt33

4 yıl öncediscuss

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).

peterhalburt33

4 yıl öncediscuss

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.

peterhalburt33

4 yıl öncediscuss

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.”

peterhalburt33

4 yıl öncediscuss

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

peterhalburt33

4 yıl öncediscuss

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.

peterhalburt33

4 yıl öncediscuss

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.

peterhalburt33

4 yıl öncediscuss

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.)

peterhalburt33

4 yıl öncediscuss

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…

peterhalburt33

4 yıl öncediscuss

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.

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