Timeline for Why have mathematicians used differential equations to model nature instead of difference equations
Current License: CC BY-SA 3.0
34 events
| when toggle format | what | by | license | comment | |
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| Oct 1, 2014 at 18:51 | answer | added | ex0du5 | timeline score: 1 | |
| Oct 1, 2014 at 8:24 | comment | added | Asaf Karagila♦ | @KConrad: The effects of not having an order is that having $-1$ balance in your bank account means that you're the richest man in the universe. | |
| Oct 1, 2014 at 3:09 | comment | added | Craig Feinstein | I didn't choose it arbitrarily. I used Occam's razor. Continuity is much more complicated conceptually than discreteness. | |
| Oct 1, 2014 at 1:33 | comment | added | S. Carnahan♦ | In the absence of solid evidence of discretized space and time, it might be best to remain undecided about the fundamental structure. Suppose that at some point we find that both a discrete and a continuous model produce good Theories of Everything of roughly equal complexity, and that it is impossible to sort out which is correct with experiments. Wouldn't that be a good reason to say that both models accurately reflect reality, rather than choosing one arbitrarily? | |
| Sep 30, 2014 at 20:38 | history | made wiki | Post Made Community Wiki by Todd Trimble | ||
| Sep 30, 2014 at 16:45 | answer | added | Doron Zeilberger | timeline score: 11 | |
| Sep 30, 2014 at 16:26 | comment | added | Craig Feinstein | I think nature is discrete. Cartoons convinced me of this. Cartoons are discrete but look continuous. Why not nature? | |
| Sep 30, 2014 at 16:21 | vote | accept | Craig Feinstein | ||
| Sep 30, 2014 at 16:21 | comment | added | Craig Feinstein | Everyone gave great answers. | |
| Sep 30, 2014 at 14:28 | answer | added | Alexey Ustinov | timeline score: 2 | |
| Sep 30, 2014 at 14:08 | comment | added | Nathan Reading | My intuition about the continuum that is (apparently) around me is just as solid as my intuition about the concept of (finite) number. The epsilon-delta construct is nothing more than a way to rigorize the fundamentally intuitive notion of limit within an axiom system built on number. (Aside: A simpler reason that we model nature by a continuum is because it has been tremendously successful!) | |
| Sep 30, 2014 at 14:08 | comment | added | Nathan Reading | The purpose of rigor in mathematics is to provide a framework for reasoning about "concepts." The most fundamental concepts cannot themselves be defined rigorously, hence the quotes around "concepts" above. Instead, the concepts come from our intuition. That's why at the base of our mathematical reasoning lie axioms that we can't prove. The purpose of the axioms is to place our concepts into the rigorous framework. SO, to the question itself: We model nature as a continuum because (despite potential small-scale quantization) we experience nature as a continuum. [To be continued] | |
| Sep 30, 2014 at 13:52 | answer | added | André Henriques | timeline score: 21 | |
| Sep 30, 2014 at 11:40 | comment | added | KConrad | effects of working with finite fields in place of $\mathbf R$, such as there being no meaningful notion of ordering: every number is "negative", in the sense of being the additive inverse of a suitable sum of 1's, many "negative" integers can be perfect squares (in $\mathbf Z/11\mathbf Z$, $-2 = 9$ so $-2$ is a perfect square), and the multiplicative inverse of a "positive" number can be "larger" than it, e.g., in $\mathbf Z/11\mathbf Z$ the inverse of 2 is 6. In reality, does half a meter seem to be larger than 2 meters? This would be completely at odds with how measurements work in reality. | |
| Sep 30, 2014 at 11:33 | comment | added | KConrad | @CraigFeinstein, the idea that there "is" a highest number, which when added to 1 makes 0, makes me wonder: what do you mean by "is"? Do you think the construction of $\mathbf R$ within pure math has a flaw, do you think counting in nature can be modeled more accurately using $\mathbf Z/n\mathbf Z$ for a mysterious large number $n$ (and if so, provide evidence), or something else? I have read one of Zeilberger's rants that we should model reality with $\mathbf Z/p\mathbf Z$ in place of $\mathbf R$ for some huge prime $p$ (to make it a field), but this ignores profound algebraic [contd.] | |
| Sep 30, 2014 at 9:12 | answer | added | Paul Siegel | timeline score: 33 | |
| Sep 30, 2014 at 5:30 | comment | added | abo | "(That highest number plus one is zero.)" This has always struck me as a curious view. What is gained by supposing the highest number plus one is zero, rather than supposing the highest number plus one does not exist? Indeed, what is gained by supposing there is a highest number, rather than just being agnostic about the matter (and just not assuming the axiom "every number has a successor")? | |
| Sep 30, 2014 at 4:56 | answer | added | Daniel Moskovich | timeline score: 14 | |
| Sep 30, 2014 at 4:10 | comment | added | Count Iblis | The q-derivative en.wikipedia.org/wiki/Q-derivative is easier to handle than finite differences, you can easily generalize the results of ordinary calculus to the finite q case | |
| Sep 30, 2014 at 4:07 | comment | added | Robert Israel | For numerical computation involving Navier-Stokes, you might indeed use some discretization. But the discretization arises as an approximation to the continuum equations, and you try to make an intelligent choice based on the properties of the continuum model. Things might not be so simple if you started from a particular set of difference equations. | |
| Sep 30, 2014 at 3:46 | comment | added | user21349 | Furthermore, difference equations don't require complicated epsilon-delta definitions. Nor do differential equations require epsilon-delta definitions. Newton and Leibniz solved differential equations about 150 years before epsilontics came along. | |
| Sep 29, 2014 at 19:53 | comment | added | Joonas Ilmavirta | @CraigFeinstein, I don't know if infinity exists. But there are some very big numbers, and it makes life (mathematics) easier to replace $10^{20}$ with $\infty$ (in a controlled way). | |
| Sep 29, 2014 at 19:45 | comment | added | Craig Feinstein | My question is motivated by my belief that there is no such thing as infinity, that there is a highest number. (That highest number plus one is zero.) But still, there might be a practical reason for using differential equations. | |
| Sep 29, 2014 at 19:24 | comment | added | Suvrit | this seems to be "re-igniting" the wave-particle debate, but now in mathematical terms :-) | |
| Sep 29, 2014 at 19:21 | comment | added | Craig Feinstein | That's true, but the Navier-Stokes equations are infinitely easier to deal with as difference equations than partial differential equations. | |
| Sep 29, 2014 at 19:00 | comment | added | Robert Israel | Case in point: you're much more likely to find closed-form solutions to the differential equation $y' = f(y)$ than to the difference equation $y(n+1) = y(n) + f(y(n))$. | |
| Sep 29, 2014 at 18:33 | comment | added | Yemon Choi | The 2nd sentence in your previous comment does not seem relevant to the 1st sentence, and the 1st sentence still seems debatable. Recall that the point of having a difference equation is to find solutions. How is that simpler than finding a solution of the corresponding differential equation for the continuum limit? Just because the definition is simpler this does not make them easier to work with | |
| Sep 29, 2014 at 18:21 | answer | added | Joonas Ilmavirta | timeline score: 49 | |
| Sep 29, 2014 at 18:02 | comment | added | Craig Feinstein | Difference equations are simpler to work with in that they are easier to understand. Furthermore, there is no evidence that continuity and smoothness are real phenomena and not just products of the senses. | |
| Sep 29, 2014 at 17:27 | answer | added | Gottfried William | timeline score: 8 | |
| Sep 29, 2014 at 17:20 | comment | added | Yemon Choi | That said, I think there could be good answers to this question; I merely wanted to point out that the starting thesis of your question is open to some debate | |
| Sep 29, 2014 at 17:19 | comment | added | Yemon Choi | Erm, because difference equations are not simpler to actually work with? What is your evidence for saying "they could have been just as successful in modeling natural phenomena with difference equations instead of differential equations" | |
| Sep 29, 2014 at 17:14 | history | edited | Craig Feinstein | CC BY-SA 3.0 |
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| Sep 29, 2014 at 17:04 | history | asked | Craig Feinstein | CC BY-SA 3.0 |