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Harvard 1869 entrance exam

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Re: Harvard 1869 entrance exam

#91

Post by gbaygon »

dlss wrote: post #66
barrkel wrote: post #58

Earlier quoted context omitted.

I don't think that's the case. As baddox says, it's trivial to show it is true, especially using simplified definitions for exponentiation (i.e. sticking with integer or perhaps rational exponents), but demonstrating truth doesn't tell you about the "reason". Is the question about a philosophical position as to how mathematics relates to God? A "reason" seems to imply a purpose.

Being able to explain proofs intuitively is a valuable way to check how deeply you know them. In this case, the reason that when different powers of the same quantity are multiplied together their exponents are added is because powers are short hand for a series of multiplications: 2^4 == 2 * 2 * 2 * 2 When you multiply 2^4 * 2^4, that is short hand for: 2^4 * 2^4 == 2 * 2 * 2 * 2 * 2 * 2 * 2 * 2 == 2^8 Of course you…

Well explained, but 2^4 * 2^4 doesn't take into account the "different powers" part.

Re: Harvard 1869 entrance exam

#92

Post by dlss »

baddox wrote: post #71
dlss wrote: post #66

Earlier quoted context omitted.

Being able to explain proofs intuitively is a valuable way to check how deeply you know them. In this case, the reason that when different powers of the same quantity are multiplied together their exponents are added is because powers are short hand for a series of multiplications: 2^4 == 2 * 2 * 2 * 2 When you multiply 2^4 * 2^4, that is short hand for: 2^4 * 2^4 == 2 * 2 * 2 * 2 * 2 * 2 * 2 * 2 == 2^8 Of course you…

That works for positive integer bases and exponents, but try giving an "intuitive proof" with irrational exponents. Most things in math, even seemingly obvious things in arithmetic, require a lot of shared background knowledge (at least propositional logic, basic set theory, and a construction of the natural numbers) for two people to even converse formally.

It still works with irrational exponents (start with fractional numbers and work towards that). It also works with imaginary exponents. It works because the power notation is short hand. But that wasn't the point of my reply. The point was that stating things multiple ways assists us in understanding. Does this not match your experience?

Re: Harvard 1869 entrance exam

#94

Post by Queue29 »

jerf wrote: post #56
baddox wrote: post #38

Earlier quoted context omitted.

That's not very rigorous, especially with fractional (or God forbid, irrational) exponents.

Given that the exam is from 1869 for entrance into college, I suspect that's basically what they are looking for.

Yes, because irrational numbers wouldn't be discovered until the 1900's....

Re: Harvard 1869 entrance exam

#95

Post by Peaker »

wbhart wrote: post #77
baddox wrote: post #70

Earlier quoted context omitted.

I would have to brush up a bit, but when I was in school I wouldn't have had much trouble proving it. But that's not the real issue. My main problem is the vagueness in wording (which might be attributable to the lack of formalization in mathematics in 1869). What does "reason" mean? Is it asking for a proof? And if so, what axioms and lemmas are you allowed to use? Are we talking about integer bases and exponents (t…

Raising a base to a power is a prescription for how many times to multiply by the base. If you first raise it to one exponent, m say, then to another exponent, n say, and then multiply, you have first multiplied by the base m times, then multiplied by the same base a further n times. In total you have multiplied by the base m + n times. Back in those days they would have used slide rules and understood logarithms ver…

That explanation is somewhat problematic with fractional exponents.

The generalization of exponentiation makes the "multiply N times" explanation fail.

Re: Harvard 1869 entrance exam

#96

Post by sneak »

jerf wrote: post #64
boredguy8 wrote: post #28

Earlier quoted context omitted.

Uhm, what? You couldn't be more off. Traditionally, any liberal arts education would include extensive familiarity with the classics. That means Plato & Dante, at the minimum, whether a BA or BS. You were expected to know Latin and Greek because you were expected to read Latin and Greek. If you were pursuing a BS, then perhaps you would read Euclid's "Elements" instead of Thucydides' "The History of the Peloponnesian…

It isn't "chronological hubris" that the vast bulk of human knowledge has been generated since 1869. Precise measurement is difficult to even define but it's difficult to imagine a non-pathological definition for which that would not be true. Personally I'm a huge advocate for the fact that what one might call "wisdom" is not unique to our age and may indeed be getting a bit lost in the shuffle, but nevertheless, the…

The bulk of current (2011) human knowledge was set in stone a thousand years before 1869.

Re: Harvard 1869 entrance exam

#97

Post by CamperBob »

32ftpersecond wrote: post #52
dexy wrote: post #8

Amazing how curriculum focus has changed--in no small part due to the invention of the computer. These topics have almost nothing to do with what most Harvard students study today. There's been so much new knowledge generated since then...

But we've strayed away from the old knowledge (e.g. classics) -- the stuff that keeps repeating itself...

If history repeats itself, then why not study newer, more relevant iterations?

Re: Harvard 1869 entrance exam

#98

Post by derleth »

tylee78 wrote: post #33
xymostech wrote: post #20

Earlier quoted context omitted.

Actually from a cursory look at the Latin part, it seemed that all of the sentences and things they chose for you to translate were very specific cases of applying rules. Maybe a bit harder than some of the other stuff, but still regurgitation.

look at the bigger picture: there is a lot to be said about mastering a classical language to such a degree that you can translate these sentences back. Sure, on the surface it looks simplistic, but students with such a command of Latin would equally possess a rich knowledge in Roman and European history and culture through the process of acquiring classical Latin. Someone mentioned "modern history" missing...well, m…

> Someone mentioned "modern history" missing.

That would be me. 1869 might be a bit early for Civil War history to show up, but nothing on the War of 1812 or even the Revolution? Nothing on the history of Westward Expansion?

I understand going back to the sources ("ad fontes") and education for its own sake, but I've never seen something that implies an education that is so divorced from anything of the time the people receiving it are living in.

Re: Harvard 1869 entrance exam

#99

Post by derleth »

veyron wrote: post #19
dhconnelly wrote: post #14

Earlier quoted context omitted.

That isn't true. Only the history/geography was recall. Latin translation and geometric proofs aren't regurgitation.

Plane geometry is woefully lacking in education nowadays. The situation is so sad that some of these questions are very similar to the geometry questions in the USAMO/IMO

Plane geometry was, at the time, perhaps the premier formal system to be studied as such; now, we get the same concepts across using calculus (which doesn't seem to be mentioned on the exam) and set theory (which would only really take shape in the 1870s).

(The history of mathematics doesn't get so much as a mention, either, but I don't expect it to.)

Re: Harvard 1869 entrance exam

#100

Post by TeMPOraL »

wolf550e wrote: post #79
barrkel wrote: post #60

Earlier quoted context omitted.

I don't think any of that is difficult ; it is, rather, a test of memorization, not ability. A lot of the mathematics section is also on the basis of memorization.

In those days, being an educated person meant memorizing a lot of things, because you couldn't haul your library with you everywhere you went, and in any case it didn't have full text search. Separate a modern educated person from google/wikipedia/etc. and see whether they can still converse at the same lavel. Most can't.

> Separate a modern educated person from google/wikipedia/etc. and see whether they can still converse at the same lavel. Most can't.

Relevant xkcd: http://xkcd.com/903/

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