I know there's something called "ultrafinitism" which is a very radical form of constructivism that I've heard said means people don't believe that really large integers actually exist. Could someone make this a little bit more precise? Are there good reasons for taking this point of view? Can you actually get math done from that perspective?
[Math] “ultrafinitism” and why do people believe it
constructive-mathematicsphilosophysoft-question
Related Solutions
You've gotten some good advice so far; permit me to chime in with the perspective of someone on the other side of the lectern. I've been teaching at small liberal arts colleges for longer (far longer) than any of my current students have been alive and over the years my colleagues and I have seen many of our majors accepted at good-quality graduate schools, both for Master's and Ph.D. programs. Let me get your major question out of the way immediately
[I]f you go to a small school, are you doomed to fail right from the get-go?
This is easy to answer: no! as I've mentioned above.
Now that that's out of the way, I first need to back up a little and ask you the question you should ask yourself,
Why do I want to get a Ph.D.?
The process of obtaining a Doctorate is long and, frankly, painful. If you've decided put your life on hold for five to seven hard and exasperating years, you'd better have a good reason. If your goal is to continue doing mathematics and eventually producing results of your own that add to the body of what is known, that's commendable. If you view the Ph.D. as a high-falutin' teaching certificate and see yourself as primarily a college professor, that's also commendable. If, on the other hand, you're contemplating merely drifting into a way to keep doing what you're good at, it might be a good idea to ask yourself whether there might be some other way to spend the next few years, since a Doctorate in mathematics won't help you very much in the Real World outside of academia and may actually harm more than help in eventually landing a job out there.
For the purpose of discussion, let's assume that you have thought long and hard and decided, "Yup, I really want to get that diploma," how can you get your foot in the door? You've earned superb grades in what looks to me to be a reasonably solid undergraduate curriculum. For all but the applicants for the very top schools, that will work in your favor, though you're correct in your assessment that good grades will be a small part of the admissions committee's view of you. Similarly, the GRE will also be a small part of your total package---those scores will primarily be a check for the Committee to ensure that you're not a hopeless idiot. On the other hand, you probably won't have any Fields medalists among your letters of reference, but that's not as bad as it may seem, especially if any recent graduates from your college have made it into a school for which you're applying.
Reference letters are the single most important part of your package, hands down. No matter where you did your undergraduate work and no matter who's writing your letters, the sentence "X was the best student I've seen in ten years" will loom large in the eyes of the Committee. If your English is excellent, make sure one or more of your letters mentions that, since universities are always on the lookout for good teaching assistant material.
Is there an area of math that really excites you? If so, check for universities with strong programs in that area and mention that in your personal statement. It's already been mentioned that it might be a good idea to write to faculty members at your target school and ask about what they're doing in the area of your interest before you submit your application.
That said, where should you apply? I agree that a top-tier graduate school is probably a stretch, though that shouldn't rule out trying for one or two. Who knows, you might get lucky? On the other hand, you probably shouldn't limit yourself to universities whose motto on their seal is "Plenty of Free Parking." There are plenty of respectable universities between those two extremes and for the time being they still need a crop of good applicants (which you are). As I said, we've sent a lot of students off to grad schools and I can't recall a case where one of our students failed to get into at least one of the schools to which they applied. Your task over the next few months is to research those middle-tier schools that look like the best fit for you.
As Ragib and Francis mentioned, getting a Master's degree first isn't all that bad an idea. At least, it might give you an idea about whether going on for the Doctorate is what you want to do (that's what I did, though for different reasons). If you go that route, do your best to make yourself stand out from the crowd, so your subsequent letters will reflect that. Keep in mind, though, that you'll have to dig up the money to pay for your education, since graduate schools rarely offer financial support for Master's students (though many will provide full support for Ph.D. students).
Finally, I fully agree with some of the other answers: don't be discouraged. Things are nowhere near as bad as you've painted them. If that's what you want, you'll almost certainly succeed. Best of luck---keep us posted.
(Sorry about the generation error!)
To attempt to answer the actual question, I think that it’s a largely two-fold consequence of the historical inertia mentioned in the comments by @coffeemath. On the one hand, it’s simple classroom inertia: it’s ‘always’ been done this way, so we do it this way. On the other hand it’s the practical consideration that since the radical notation does survive in real-world use, students need to learn how to deal with it. None of this, however, justifies the requirement that students deal with it directly, rather than by translating it into a less cumbersome, more easily manipulated notation. Indeed, in my view this is a good occasion to make the point that well-chosen notation makes our mathematical lives easier.
Best Answer
Ultrafinitism is basically resource-bounded constructivism: proofs have constructive content, and what you get out of these constructions isn't much more than you put in.
Looking at the universal and existential quantifier should help clarify things. Constructively, a universally quantified sentence means that if I am given a parameter, I can construct something that satisfies the quantified predicate. Ultrafinitistically, the thing you give back won't be much bigger: typically there will be a polynomial bound on the size of what you get back.
For existentially quantified statements, the constructive content is a pair of the value of the parameter, and the construction that satisfies the predicate. Here the resource is the size of the proof: the size of the parameter and construction will be related to the size of the proof of the existential.
Typically, addition and multiplication are total functions, but exponentiation is not. Self-verifying theories are more extreme: addition is total in the strongest of these theories, but multiplication cannot be. So the resource bound is linear for these theories, not polynomial.
A foundational problem with ultrafinitism is that there aren't nice ultrafinitist logics that support an attractive formulae-as-types correspondence in the way that intuitionistic logic does. This makes ultrafinitism a less comfy kind of constructivism than intuitionism.
Why do people believe it? For the same kinds of reasons people believe in constructivism: they want mathematical claims to be backed up by something they can regard as concrete. Just as an intuitionist might be bothered by the idea of cutting a ball into non-measurable pieces and putting them back together into two balls, so too an ultrafinitist might be concerned about the idea that towers of exponentials are meaningful ways of constructing numbers. Wittgenstein argued this point in his "Lectures on the Foundations of Mathematics".
Can you actually get math done from that perspective? Yes. If intuitionism is the mathematics of the computable, ultrafinitism is the mathematics of the feasibly computable. But the difference in ease of working with between ultrafinitism and intuitionism is much bigger than that between intuitionism and classical mathematics.