jfarmer

37 ms·

301.
by jfarmer 13y ago
I typed it from memory, so I could be wrong. I thought the 13th axiom was the least upper bound property.
302.
by jfarmer 13y ago
If you took me to mean the "default" first-year calculus textbook then yes, that's not common. But it's definitely aimed at first-year college students, or at least people who haven't had prior exposure to rigorous
303.
by jfarmer 13y ago
Indeed! Me and my fat fingers. :)
304.
by jfarmer 13y ago
Spivak's Calculus starts with a set of 13 axioms which characterize the real numbers and then derives all the results you're familiar with in calculus. It's rigorous in the mathematical sense, so if you've never worke
305.
by jfarmer 13y ago
What are you looking for that isn't simply a math textbook? Honestly, that's how mathematicians learn "actual" mathematical subjects, too. If you're comfortable with calculus as a subject, for example, and want a &
306.
by jfarmer 13y ago
Polya's How to Solve It was recommended below. I also like How to Prove it by Daniel Velleman ( http://www.amazon.com/How-Prove-It-Structured-Approach/dp/05... ).
307.
by jfarmer 13y ago
Well, we can and have made the idea of a computable number precise: http://en.wikipedia.org/wiki/Computable_number It's not clear whether the universe is computable, however, in the sense that we only find computa
308.
by jfarmer 13y ago
The trick here is defining "occur" and "real world" precisely. Are you saying the act of me writing down those symbols and expressing the idea does not count as "occurring in the real world?" ;)
309.
by jfarmer 13y ago
Well, "in some sense" gives a lot of wiggle room! I agree the magnifying glass analogy is bad, but here's one "sense" in which the rationals "have more holes" than the irrationals. There exists a functi
310.
by jfarmer 13y ago
You're using too many imprecise terms. First, what does it mean for something to be able to be "unambiguously" described? As opposed to ambiguously described? You have to define it. Second, what does it mean for something
311.
by jfarmer 13y ago
Conversely, I find infinitessimals vague and woo-woo, especially the way physicists and statisticians often use them. Once the epsilon-delta style "clicked" for me, it felt like second nature. How can you tell whether it's s
312.
by jfarmer 13y ago
Sorry, but the semantics of "this" in JavaScript are crazy — and I think that's being generous. There are a few common use cases that trip people up, e.g., situations where "this" becomes bound to another object li
313.
by jfarmer 13y ago
Earlier this year I implemented the arithmetic of ℚ[φ] in Ruby, to illustrate how something like this would work for my students: http://bit.ly/1gSQdpC Obviously the Haskell implementation more directly expresses the idea!
314.
by jfarmer 13y ago
To answer the question, one has to define precisely what "returning multiple values" should mean. It's hard to see how this is indistinguishable from returning a list/array/tuple plus supporting list destructuring s
315.
by jfarmer 13y ago
Yes. Although I disagree that what the GP rebuts what he was intended to rebut, I agree with his particular point, viz., hat the idea of OOP doesn't necessarily entail the idea of classes, although most OOP languages implement many pr
316.
by jfarmer 13y ago
Whether those questions come out of a student's mouth, those questions will be rattling around their head — most likely much fuzzier, more confused version of those questions. If you thought I was proposing that a teacher, say, give a
317.
by jfarmer 13y ago
Think about all the personal and professional feedback you've received over your entire life. Have you ever been told you're too curt, abrasive, or aggressive? Has anyone told you that what you say is fine, but how you say i
318.
by jfarmer 13y ago
Those are the concepts starting the beginner in the face, sure. Here some of the questions a beginner will have, phrased ~10x more clearly than a beginner would ever be able to phrase them: 1. What is an object? 2. What does it m
319.
by jfarmer 13y ago
Your point is, well, beside the point. You're saying that "OOP does not imply static methods," which is correct, but irrelevant, because the converse is true: "Static methods (in the Java sense) imply OOP." Assumi
320.
by jfarmer 13y ago
My educated guess is that, broadly speaking, there are three classes of people reading something on Casetext: (1) the people raising, (2) investors, and (3) the attorneys advising them. Let's assume for now that all three groups rough
321.
by jfarmer 13y ago
> it is an educational tool so one would expect one to read the getting started pages first This is the opposite of how it should be. You'd expect learners to be less likely to read a "getting started" page than an expert
322.
by jfarmer 13y ago
This boundary: http://en.wikipedia.org/wiki/Cosmological_horizon
323.
by jfarmer 13y ago
Well, it's not an analogy! There's a 1-1 correspondence between matrices and linear maps over a vector space with a fixed basis. "A matrix represents a function" is no more an analogy than "The symbols 'V&#x2
324.
by jfarmer 13y ago
First, if you can handle it, I recommend the original paper proposing monads as a solution for a wide variety of problems that every functional language has to face: http://homepages.inf.ed.ac.uk/wadler/papers/mark
325.
by jfarmer 13y ago
These arguments are specious. Most Libertarian proponents of a guaranteed income don't favor it because it will "save money," but because it's the least economically distortive way of redistributing wealth. The argument
326.
by jfarmer 13y ago
Convertible notes are common across all companies raising early (seed) money, even ones outside YC. I don't know the landscape now, but the last time I was involved in raising money (2011 Q2), convertible notes were the presumptive de
327.
by jfarmer 13y ago
This article is confused and its confusion is apparent right from the first paragraph: "In the course of exploring their universe, mathematicians have occasionally stumbled across holes: statements that can be neither proved nor refute
328.
by jfarmer 13y ago
The three basic (metaphysical) axioms according to Objectivists are Existence, Identity and Consciousness. Anyhow, as far as I can tell, all you're saying is that either (1) Cantor's argument is invalid within ZF(C) or (2) you fi
329.
by jfarmer 13y ago
You out yourself by using the phrase "Reification of Zero," which is something I've only ever heard from Objectivists. It's funny to hear you say this, since Objectivists are usually so apoplectic about axioms and premi
330.
by jfarmer 13y ago
Not quite. I can represent pi in a finite way, like so: π. That's about as finite as it gets! However, if you fix a finite alphabet beforehand, you will never be able to describe all real numbers using strings constructed from this a

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