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The scenario in the OP is not very promising: E.g., we can study the natural numbers and the integers, rationals, algebraics, and reals, but why the reals? We

by graycat 12y ago

The scenario in the OP is not very promising: E.g., we can study the natural numbers and the integers, rationals, algebraics, and reals, but why the reals? Well, they are the only complete, Archimedean ordered field. Why do we care? We want the order because we want to consider bigger and smaller. We want a field because we want to be able to do basic arithmetic. But why do we want completeness? Because it says that pi, e, square root of 2, derivatives, and the Riemann integral exist with few or no more additional assumptions. Why? With completeness we get compactness, continuity, and uniform continuity and, thus, know that the Riemann integral of a real valued continuous function on a closed interval of finite length exists and we also get the fundamental theorem of calculus. Why do we care about calculus? Because it is the main way to discuss, analyze, and understand continuous change, e.g., Newton's second law and planetary motion, fluid flow, heat flow, electro-magnetism, etc. And why do we care about such analysis? Because it has been one of the pillars of physical science, engineering, the industrial revolution, and the growth of technology and economic productivity since Newton -- that is, we care about applications. So, to understand mathematics and why we do it and why some math is important, need also to understand applications to the betterment of human life on this planet, and I'm not holding my breath while waiting for the automation of that! Apparently now there is a big theme, climb on the bandwagon, ride the wave along with everyone else trying to get people up on their hind legs to grab their eyeballs for ad revenue, of with artificial intelligence, the machines are about to take over. We've had many waves before -- they come and they go. And we've had that wave of the computers about to take over before, and apparently we will more times before computers can get even remotely close to any such thing. Ah, basic rule: "Always look for the hidden agenda.". Here, it feels to me like some people are grabbing at me, by the heart, the gut, maybe below the belt, definitely below the shoulders and not between the ears.

BrainInAJar 12y ago
> E.g., we can study the natural numbers and the integers, rationals, algebraics, and reals, but why the reals? Well, they are the only complete, Archimedean ordered field. Well, and the complex numbers, they're the only algebraically closed field.
- omaranto 12y ago
  > Well, and the complex numbers, they're the only algebraically closed field. That is very, very far from being true! There are algebraically closed fields of every characteristic and of arbitrarily large cardinality. What did you mean?
  - deleted 12y ago
    [deleted]
  - BrainInAJar 12y ago
    sorry, "the only" was way wrong. They are, conveniently, an algebraically closed field.
  - JadeNB 12y ago
    > > Well, and the complex numbers, they're the only algebraically closed field. > That is very, very far from being true! There are algebraically closed fields of every characteristic and of arbitrarily large cardinality. What did you mean? Although BrainInAJar seems simply to have misspoken (https://news.ycombinator.com/item?id=9254999 https://news.ycombinator.com/item?id=9254999), you have just explained one of the things that the remark could have meant: namely, that ℂ is the unique characteristic-0, algebraically closed field with the cardinality of the continuum.
bjwbell 12y ago
Sometimes I wonder if the usefulness of calculus is overstated compared to other areas of math. Linear algebra & discrete math pop up a lot more than integrals in CS. I loved my real & complex analysis courses but my courses on linear algebra, abstract algebra, & discrete math have been much more useful.
- graycat 12y ago
  I can understand and essentially agree with everything you said except your first sentence, and I'm not sure the rest of what you said does much to support your first sentence! One point is, CS isn't the only area for applications! E.g., in the software for my startup, sure, I have some matrix theory, right there in the code, but it turns out the matrix theory is what is left for the actual code after some earlier derivations very much in calculus! When I was a prof in a B-school and teaching linear programming, right, awash in linear algebra, I mentioned to my students, all of whom had had the required courses in calculus, that I regarded it as a "pillar of Western Civilization". I still do. Without Newton's second law, Maxwell's equations, etc., I strongly suspect that Western Civilization would be a very different and much less good place. E.g., my father in law eventually slowed down his farming and got a job in town. He was head of the REMC -- Rural Electric Membership Cooperative. So, it was the local electric utility. They handled only the last few miles and bought their electric power from the grid, really from one private power company. Some of his customers were factories, and at one point he asked me why his engineers put large capacitors outside some of the factories. Well, I'd been a ugrad math major but, except for one course I wanted instead of another that would have been required, also a physics major, and had done well in ordinary differential equations, so had see the differential equations of basic passive AC circuit theory, that is, with resistors, capacitors, and inductors. So, sure, the factories had a lot of big electric motors with a lot of inductance. So, the utility pushed current to the motors but half a cycle later the motor pulled more current. So, net, the utility was moving a lot more electrons than necessary to deliver the power it was getting paid for and, thus, was getting more power losses in its lines. So, put a capacitor just outside the plant, and then the plant looks like a pure resistor to the electric company and all the extra electron moving is just between the motors and the capacitor just outside the plant. Ah, applied calculus! There are many more such examples; the examples say that calculus is really important but don't really settle your question about "overstated"; for that question, I don't know what to say!

by graycat 12y ago

BrainInAJar 12y ago

> E.g., we can study the natural numbers and the integers, rationals, algebraics, and reals, but why the reals? Well, they are the only complete, Archimedean ordered field. Well, and the complex numbers, they're the only algebraically closed field.

omaranto 12y ago
> Well, and the complex numbers, they're the only algebraically closed field. That is very, very far from being true! There are algebraically closed fields of every characteristic and of arbitrarily large cardinality. What did you mean?
- deleted 12y ago
  [deleted]
- BrainInAJar 12y ago
  sorry, "the only" was way wrong. They are, conveniently, an algebraically closed field.
- JadeNB 12y ago
  > > Well, and the complex numbers, they're the only algebraically closed field. > That is very, very far from being true! There are algebraically closed fields of every characteristic and of arbitrarily large cardinality. What did you mean? Although BrainInAJar seems simply to have misspoken (https://news.ycombinator.com/item?id=9254999 https://news.ycombinator.com/item?id=9254999), you have just explained one of the things that the remark could have meant: namely, that ℂ is the unique characteristic-0, algebraically closed field with the cardinality of the continuum.

bjwbell 12y ago

Sometimes I wonder if the usefulness of calculus is overstated compared to other areas of math. Linear algebra & discrete math pop up a lot more than integrals in CS. I loved my real & complex analysis courses but my courses on linear algebra, abstract algebra, & discrete math have been much more useful.

graycat 12y ago
I can understand and essentially agree with everything you said except your first sentence, and I'm not sure the rest of what you said does much to support your first sentence! One point is, CS isn't the only area for applications! E.g., in the software for my startup, sure, I have some matrix theory, right there in the code, but it turns out the matrix theory is what is left for the actual code after some earlier derivations very much in calculus! When I was a prof in a B-school and teaching linear programming, right, awash in linear algebra, I mentioned to my students, all of whom had had the required courses in calculus, that I regarded it as a "pillar of Western Civilization". I still do. Without Newton's second law, Maxwell's equations, etc., I strongly suspect that Western Civilization would be a very different and much less good place. E.g., my father in law eventually slowed down his farming and got a job in town. He was head of the REMC -- Rural Electric Membership Cooperative. So, it was the local electric utility. They handled only the last few miles and bought their electric power from the grid, really from one private power company. Some of his customers were factories, and at one point he asked me why his engineers put large capacitors outside some of the factories. Well, I'd been a ugrad math major but, except for one course I wanted instead of another that would have been required, also a physics major, and had done well in ordinary differential equations, so had see the differential equations of basic passive AC circuit theory, that is, with resistors, capacitors, and inductors. So, sure, the factories had a lot of big electric motors with a lot of inductance. So, the utility pushed current to the motors but half a cycle later the motor pulled more current. So, net, the utility was moving a lot more electrons than necessary to deliver the power it was getting paid for and, thus, was getting more power losses in its lines. So, put a capacitor just outside the plant, and then the plant looks like a pure resistor to the electric company and all the extra electron moving is just between the motors and the capacitor just outside the plant. Ah, applied calculus! There are many more such examples; the examples say that calculus is really important but don't really settle your question about "overstated"; for that question, I don't know what to say!