Timeline for Why do we need geometry for pure math?
Current License: CC BY-SA 4.0
13 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Aug 29, 2019 at 3:24 | comment | added | user9166 | The past century of math has also given us plenty of math without numbers from topology to cartesian categories to the classification of finite simple groups. So how is it that they are most definitely necessary? Yet, if I asked why anyone needs them, you could definitely see that as an adverse agenda. And you wouldn't be projecting. Nor am I. | |
| Aug 28, 2019 at 23:01 | comment | added | Justin Latson | To ignore the past century of the history of math is bigotry. I am not in any way saying one way is superior or not. This is something you project. I simply ask, why is geometry necessary. And you can't provide an answer. Algebra and numbers are most definitely necessary to math. | |
| Aug 28, 2019 at 22:20 | comment | added | user9166 | I am a constructivist (or at least an intuitionist). For me, approximation is more often a solution than the airy hand-waving behind more 'rigorous' math if it involves non-constructive existence. But you want to dictate the opposite. There is no real reason behind your perspective. Mine at least makes computers happy. | |
| Aug 28, 2019 at 22:17 | comment | added | user9166 | This has gone on too far. I think your whole point here is bigotry against what does not appeal to you, and you are not open to that. But the fact one can or might scrape the geometry off calculus does not mean one should just so folks who prefer computation feel safer. Who cares whether we have seven different ways of looking at things, as long as they help? Well, only folks who would rather have control that discoveries and who disapprove of others' methods: In other words impractical bigots. | |
| Aug 28, 2019 at 22:13 | comment | added | user9166 | Chemists in my generation (prior to calculators) definitely imagined calculus with no operations. They did derivatives by interpolating slopes and integrals by graphing functions on filter paper, cutting out the area and weighing it. The idea for that is inspired by the geometric model of the discipline, and has nothing to do with its algebra. That perspective is a gift from math to chemistry. | |
| Aug 28, 2019 at 21:07 | comment | added | Justin Latson | Could you imagine calculus with no numbers or operations? Could you imagine a calculus with only shapes? Because I can see a calculus without geometry. But vice versa I doubt it'd be very communicable. | |
| Aug 28, 2019 at 21:05 | comment | added | Justin Latson | And that's exactly my point. Approximation is not a solution. It is a mere shadow in Plato's cave to the real forms behind them. We can only approximate better and better with less of a margin of error the more times we iterate a certain computational identity. We will, however, never attain it. I also disagree that real analysis is not based on algebra. How would a power series look without the operations or the integers? Trigonometric identities can be formulated by algebraic means as well. Hey are simply one in an infinite number of combinatorical logic. | |
| Aug 28, 2019 at 20:47 | comment | added | user9166 | And (maybe unique among my posts on math) I did not mention intuition. I mentioned derivation and exposition of results. Employing more means and a greater range of sensory models facilitates derivation, and any kind of exposition that we can find to be well-grounded is useful. No one of them is really better than any other. Detecting when an algebraic proof is wrong is really not easier than detecting when one assisted by geometry is wrong. | |
| Aug 28, 2019 at 20:39 | comment | added | user9166 | Notoriously, finding the root of an arbitrary fifth dimensional polynomial is possible only if you bring in real analysis or trigonometry -- both are based in geometry, algebra does not, in itself, allow for arbitrarily close approximations and the fifth degree does not have a closed form, because of Abel's Theorem. | |
| Aug 28, 2019 at 20:09 | comment | added | Justin Latson | I just have to add one thing. Intuition has a history of deceiving us. The shock that mathematicians felt when they learned that not all continuous functions have a derivative is one example I'd cite. Although Poincare claimed that the source of the continuum is geometry. And therefore all math is essentially geometric seeing as to how infinity is essential to math. O still feel as if this is not necessarily true. Name one problem that algebra can't solve but geometry can. Or, heck, name a concept. I can name you multiple that geometry can't. So can history. | |
| Aug 28, 2019 at 14:29 | history | edited | user9166 | CC BY-SA 4.0 |
edited body
|
| Aug 28, 2019 at 12:38 | history | edited | user9166 | CC BY-SA 4.0 |
added 162 characters in body
|
| Aug 28, 2019 at 12:32 | history | answered | user9166 | CC BY-SA 4.0 |