The ACM disagrees, and I am inclined to take their word over yours. From their statement on E-voting:
voting systems should enable each voter to inspect a physical (e.g., paper) record to verify that his or her vote has been accurately cast and to serve as an independent check on the result produced and stored by the system.
For instance, both Newton and Leibinitz incorrectly used infintesimals in their definitions.
That was the old view. There were some problems with their use of infinitesimals, but those problems have been cleared up more recently. The modern version of calculus via infinitesimals is known as nonstandard analysis. The landmark work on the subject is Robinson's 1966 book "Non-standard analysis".
Moreover, that sort of hen-pecking at Newton and Leibniz is not really productive. No one cares more about precision and correctness in definitions than mathematicians, and yet mathematicians still assign credit to those two.
Have you read the Principia? I have only read portions, but Newton does some pretty amazing stuff in there, besides just the use of calculus and the derivation of the inverse square law for gravity.
For example, he proves that there is no closed form for elliptic integrals of a certain kind.
Several posters stated Abel's (or Galois'? or Ruffini's?) theorem correctly: it says that for n = 5, 6, 7, etc. there is a polynomial involving x^n whose roots cannot be expressed in terms of the four functions (+, -, *,/) and m-th roots (m = 2, 3, 4, 5...).
But that's not the whole story. Of course, if sin(1) is a root of your polynomial, then most people would be happy and consider that a perfectly good number that we can sink our teeth into. It is in fact possible that there exist polynomials of degree 5 whose roots cannot be expressed using buttons on your calculator (assuming your calculator is somehow infinite precision). That means using exponentiation, natural logs, arcsinh, etc. A particular example is the polynomial
2x^5 - 10x + 5
Well, with current math we can't prove that the roots of this polynomial have this property. But if you assume the as-yet-unproven (or as-yet-disproven, take your pick) "Schanuel's Conjecture" from number theory, you can indeed prove that this polynomial's roots are in some sense "inexpressible".
Yeah, yeah, of course you can approximate them numerically. Any australopithecine could realize that in the time it takes to gnaw an antelope femur down to the marrow.
(Personally speaking, I find the possibility that we can't explicitly write down the roots of a quintic polynomial -- especially such a nice one -- somewhat disturbing.)
Reference for the claims above: Timothy Chow, What is a closed-form number?, American Mathematical Monthly, May 1999, vol. 106, pages 440-448.
The high price of journals seems to be straight up profiteering by commerical publishers.
To follow up on what you wrote above, the entire administration of the journal is nearly free. The only place money goes is the salary of one secretary for the journal's managing editor and mailing costs for those journals that actually still mail out hardcopies to reviewers. The journal editor rarely gets any money from the journal, and the referees never do as far as I can tell. In principle, the only legitimate reason for high subscription prices is small circulation.
Looking at actual subscription prices, journals published by research societies (like the American Mathematical Society, Documenta Mathematica), university consortia (Pacific Journal of Mathematics, Annals of Math), etc. (Mathematical Research Letters), are much cheaper than those published by commercial publishers like Elsevier and Springer (Inventiones Mathematicae). The journals seem to be run the same way, so traditional publishers must be skimming profits.
But as someone employed as a sysadmin with a Liberal Arts degree, I would humbly suggest that you might think about reversing that order. Get an education first, then worry about getting job skills. An education will let you figure out what you actually want out of life; you can then decide what if any employment will help you achieve those goals.
I couldn't agree more. And here's another reason :
As anyone in college can tell you, those freshman classes seem hard to most students when they start college. But by the time most students are about to graduate, a freshman course (even in a different area) is pretty easy.
All that "learning how to think" apparently makes you much better at learning. So you can pick up those job skills much more easily after that liberal arts education.
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The ACM disagrees, and I am inclined to take their word over yours. From their statement on E-voting:
See the original statementThat was the old view. There were some problems with their use of infinitesimals, but those problems have been cleared up more recently. The modern version of calculus via infinitesimals is known as nonstandard analysis. The landmark work on the subject is Robinson's 1966 book "Non-standard analysis".
Moreover, that sort of hen-pecking at Newton and Leibniz is not really productive. No one cares more about precision and correctness in definitions than mathematicians, and yet mathematicians still assign credit to those two.
Have you read the Principia? I have only read portions, but Newton does some pretty amazing stuff in there, besides just the use of calculus and the derivation of the inverse square law for gravity. For example, he proves that there is no closed form for elliptic integrals of a certain kind.
But that's not the whole story. Of course, if sin(1) is a root of your polynomial, then most people would be happy and consider that a perfectly good number that we can sink our teeth into. It is in fact possible that there exist polynomials of degree 5 whose roots cannot be expressed using buttons on your calculator (assuming your calculator is somehow infinite precision). That means using exponentiation, natural logs, arcsinh, etc. A particular example is the polynomial
2x^5 - 10x + 5
Well, with current math we can't prove that the roots of this polynomial have this property. But if you assume the as-yet-unproven (or as-yet-disproven, take your pick) "Schanuel's Conjecture" from number theory, you can indeed prove that this polynomial's roots are in some sense "inexpressible".
Yeah, yeah, of course you can approximate them numerically. Any australopithecine could realize that in the time it takes to gnaw an antelope femur down to the marrow.
(Personally speaking, I find the possibility that we can't explicitly write down the roots of a quintic polynomial -- especially such a nice one -- somewhat disturbing.)
Reference for the claims above: Timothy Chow, What is a closed-form number?, American Mathematical Monthly, May 1999, vol. 106, pages 440-448.
To follow up on what you wrote above, the entire administration of the journal is nearly free. The only place money goes is the salary of one secretary for the journal's managing editor and mailing costs for those journals that actually still mail out hardcopies to reviewers. The journal editor rarely gets any money from the journal, and the referees never do as far as I can tell. In principle, the only legitimate reason for high subscription prices is small circulation.
Looking at actual subscription prices, journals published by research societies (like the American Mathematical Society, Documenta Mathematica), university consortia (Pacific Journal of Mathematics, Annals of Math), etc. (Mathematical Research Letters), are much cheaper than those published by commercial publishers like Elsevier and Springer (Inventiones Mathematicae). The journals seem to be run the same way, so traditional publishers must be skimming profits.
You can find data and the prices of math journal subscriptions at Rob Kirby at UC Berkeley and Ulf Rehmann at Bielefeld and John Baez at UC Riverside