Terry Tao's generals (2012) (opens in new tab)

I have tried to read the Tractatus a couple of times and whiffed majorly. I think Wittgestein foresaw Powerpoint culture and created a (the only?) major philosophical work[0] consisting of just bullet points.

[0] True, I didn't understand it but it clearly has had an impact!

> Von Neumann(an Austrian aristocrat)

Usually Von Neumann is thought to be Hungarian. Born in Budapest, which at the time was part of the Austro-Hungarian Empire.

What we today call science was once called natural philosophy. Philosophy was simply the systematic study of the world, including physics, logic, and math, as opposed to theology or learning an applied professional trade such as medicine, law, or architecture.

The terms we use to differentiate fields of study are still shifting. Nobody uses the term "naturalist" anymore, at least not to refer to a scholarly field. There's often a huge overlap between math and various other fields. I'm not sure the recent explosion in degree names is any better than using Ph.D. It's tempting to suggest that fields like Art History and English Literature should be differentiated from [modern] Physics, but ideally scholars in all those fields should be (and often are) using many of the same analytical techniques. And realistically, I doubt usage of a shared signifier, Ph.D, is causing any consequential confusion. Arguably, as specialization increases the value of using Ph.D increases--it general implies someone has attained the highest credential in their field of study, whatever the field.

In the medieval university system, there were four fields of knowledge: medicine (MD), law (Juris Doctor, or JD), theology (ThD), or philosophy (PhD).

In the centuries since, new kinds of knowledge were discovered, but they generally got squeezed in as sub-categories of philosophy, so when you specialise in them, you get a PhD.

PhD is a license to teach (i.e. doctor) philosophy. Specifically, in "science" fields, natural philosophy.

Natural philosophy was the original (and i think better) name for science, especially physics. It was understood to be an important subbranch of philosophy.

I think its better name because it elucidates an important epistemological difference: science is a method, not knowledge. I learn science when I learn about experimental design. Im not learning science when I learn about evolution; rather Im learning a theory, very likely to be true, that is almost impossible to put under scrutiny using the scientific method.

The Pythagorean theorem has no science and is 100% true. In fluids, Bernoulli devised experiments to demonstrate to his calculus illiterate colleagues what he had already mathematically proven.

Science is a bad term.

The term philosophy is used in the original Greek sense, "love of knowledge". As opposed to, Wikipedia tells me: theology, medicine and law. I guess it also has something to do with philosophy in the early sense of the word, before natural philosophy branched off and became what we today call the sciences.

The "philosophy" in PhD refers to philosophy/wisdom in general (philos=love, sophia=wisdom) not the specific academic discipline.

Partially, it’s tradition at this point, but it’s also worth pointing out that a PhD isn’t the only doctorate, see e.g. DBA, EdD, JD, MD, etc.

Philosophy is the study of everything that is not well understood or is difficult to understand, even the study of study itself. A good way to think of it is, "philosophy exists only in the gaps", the gaps in our understanding. What is the nature of matter, can you take clay and keep dividing it in half and in half? When we didn't know the answer, only speculated about it, it was philosophy. Now that we know the answer, it's physics. But unknown answers in physics are still considered philosophy, such as whether or what would it mean if time can run backward.

This implies philosophy and philosophers will disappear once we know everything, which could by some be considered one of the rewards of this quest for truth.

BTW, many institutions today offer the ScD degree, doctor of science; it is generally equivalent to a PhD, though in some places it can be considered "higher"

Early in the tradition of thinking, practitioners of thinking were called philosophers. In those times there was no distinction between the physics of the world and of the mind, physics and metaphysics were one in the same. As philosophy evolved, new branches became distinct, mathematics, physics, and later chemistry and others.

The basis of all of these branches of the sciences owe their place to the work done by the early thinkers who established the ways of knowing. It is in the method of knowing that the highest degree in academic sciences is a Doctor of Philosophy. By obtaining the degree you have proven you are capable of establishing "knowing". Which is to say, you can reason.

He took the doctorate degree in Europe. He gained a PhD in Physics.

"Doctor of Philosophy" (PhD): a doctorate in just about any subject except law or medicine.

"Doctor of Letters" (D.Lit.): a doctorate in Law.

"Doctor of Medicine" (D.Med.): a doctorate in medicine.

Philosophy literally means the love to knowledge, and was (in some way still is) the father/mother of all subjects we today call science. Before "sub-genres" like Math, Physics, Chemistry, Biology were large enough to warrant their own subjects, they were all just part of philosophy. Today, we speak of philosophy if we mean topics that are not part of these subjects, altough I feel a fair argument could be made that they could be still considered part of philosophy. The name just is something left of that time.

I don't ask questions, I just use mine for what it's worth!: Not much and decreasing by the day. (Also not in math)

Philosophy used to include all of science, up until "Natural Philosophy" got big enough to separate out on its own.

Because academics love to cling to outdated and today factually incorrect terminology, while arrogantly berating the general public when they do the same.

"Actually, it's not called the Sapir–Whorf hypothesis anymore. The preferred term is 'linguistic relativity' now."

... while you keep using titles that have been nonsensical for 200 years. Got it.

Academia is 99% power games. The emperor wears no clothes.

Yes, what you're pointing out is that it doesn't scale linearly. But the point still stands.

Typically exams are 2 full professors and maybe one asst prof. That asst prof will be 3-4 years in, plus 4-5 years of grad school + often a postdoc. And that asst prof will, on average, be better than the graduate student.

I noticed that Evan O’Dorney’s generals was chaired by Bhargava and this question was asked:

  What is Brauer's theorem?
    [I had no idea and they moved on]

He clearly should have read Bhargava’s generals, where Andrew Wiles asks Bhargava the same question!

I think his generals are easier than Tao's. I wonder how many "average" PhD candidates worldwide can answer the questions in Tao's generals satisfactorily without difficulties. Many of them just seem highly research-oriented.

that was a very entertaining one!

Yeah, my comment was a half-hearted joke. That said, the idea should work though: ask the candidate of fundamentals in their line of work, and see if the candidate can have coherent understanding of the landscape and can dive into critical details if needed.

TFA?

For sure, definitely not equating my PhD experience to Tao's. I felt like even in my viva the examiners didn't push me too hard and stuck close to the material in the thesis. Probably due to my decidedly non-prodigy status!

In life sciences the qualifying exam is an oral exam where they can in theory ask you anything. In comparison with for example the physics qualifying exam which (as I’m told) is a traditional paper exam, just ridiculously difficult.

The exams would be at the beginning to middle of the program.

> graduate students

As far as I understand, in America graduate student can mean both a M.Sc. student and a Ph.D. student.

Fascinating. I guess everyone has had that experience of "hey how does that guy never f*ck up his exams?". To an extent I was that guy in high school. In uni I wasn't, but I knew people who seemingly always got top marks. I guess everyone hits a wall at some point.

> Sometimes called "orals," "comprehensive exams," etc.

— other names I have heard people use for these: prelims, quals, ...

Fields Medal winner in 2006.

(And, as listed on his wikipedia page, 27 other awards, medals, and prizes.)

I knew that, I was wondering if this was Tao rejecting someone or being rejected himself!

I feel good about myself if I can understand more than the first sentence of any of his posts. Some days I'll even settle for only the first.

Great mathematicians are said to be exceptional at visualization. I wonder if the modern greats think in TeX or LaTeX on top of that. It is amazing how fast they post fully-formed, meticulously referenced blog articles or upload preprints.

Often the advanced theorems and concepts make for a simpler and more concise explanation.

I guess the closest equivalent in programming is "implement X without using any external libraries (or perhaps only the standard library)".

I don’t know, I keep hearing about the Tao of this, or the Tao of that.

I am sure no one has ever ever called him “The Tao of Math” before, but they have now! He is in it. Everywhere!

Legacy Achievement: locked in :)

Interesting, but my first answers are essentially - use some clever math.