tag:blogger.com,1999:blog-17232904.post4247652708734233464..comments2023-03-28T13:34:01.339+00:00Comments on The Spline: Interpreting mathematicsStephen Luttrellhttp://www.blogger.com/profile/11094835879740297834noreply@blogger.comBlogger7125tag:blogger.com,1999:blog-17232904.post-65146200680329617002008-03-31T18:49:00.000+00:002008-03-31T18:49:00.000+00:00Thanks for clearing up my confusion. I must have g...Thanks for clearing up my confusion. I must have glossed over that section of the paper...<BR/><BR/>I have a new link for you, which you <A HREF="project343.blogspot.com" REL="nofollow">may find interesting</A>.Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-17232904.post-88403011571174545842008-03-31T10:00:00.000+00:002008-03-31T10:00:00.000+00:00The original point of my blog posting (and of the ...The original point of my blog posting (and of the <A HREF="http://arxiv.org/abs/0712.1187" REL="nofollow">paper</A> it was commenting on) was that the students failed to use simple visualisation to spot an error that they had made earlier on in their calculation, i.e. to use +/- infinity as the integration limits. The students had chosen these incorrect limits, not the examiner.<BR/><BR/>You are correct to observe that the function does not have a finite integral over the +/- infinity range, and this is what the students eventually realised after far too long. In detail, the <I>actual</I> problem the students were trying to solve is an expectation value in an infinitely deep square potential well problem in quantum mechanics, for which the range of integration is physically bounded by the edges of the potential well. Realising that the range of integration must be <I>finite</I> (i.e. bounded by the edges of the infinitely deep square potential well) was merely "debugging" the students' formulation of the problem.Stephen Luttrellhttps://www.blogger.com/profile/05612770386489550404noreply@blogger.comtag:blogger.com,1999:blog-17232904.post-50481243213763886412008-03-30T19:16:00.000+00:002008-03-30T19:16:00.000+00:00Perhaps this is my problem: the question asks for ...Perhaps this is my problem: the question asks for the integral from +/- infinity. Why is "the function doesn't got a finite integral over that range" not the correct answer?Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-17232904.post-26966669608868649122008-03-30T09:45:00.000+00:002008-03-30T09:45:00.000+00:00The integral does not converge when the integratio...The integral does <I>not</I> converge when the integration limits are +/- infinity, but the students failed to recognise this fact which is obvious when you plot the function.<BR/><BR/>I wanted to point out in my blog posting that <I>visualisation</I> is a tool that you should use to get a feel for the inner workings of mathematics, including situations that are much more complicated than the simple integral here.Stephen Luttrellhttps://www.blogger.com/profile/05612770386489550404noreply@blogger.comtag:blogger.com,1999:blog-17232904.post-48193489269213105832008-03-30T03:19:00.000+00:002008-03-30T03:19:00.000+00:00Hmmm.. I struggled to understand how the integral ...Hmmm.. I struggled to understand how the integral could conere from reading your description, so I read the paper. Am I right in saying that the choice of constant to multiply by in sin ^ 2 is releant? Does multplying by pi n make the sin ^ 2 conerge faster than the ^ 2 dierges?<BR/><BR/>(sorry, broken '' and '' keys)Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-17232904.post-21414676133541763582008-02-03T19:44:00.000+00:002008-02-03T19:44:00.000+00:00I agree with your comment about lack of visualisat...I agree with your comment about lack of visualisation in <I>both</I> conventional and symbolic algebra cases, but the tendency to focus on calculations alone when you have access to symbolic algebra means that visualisation gets less of a chance to be used unless you actively develop the habit of doing it.<BR/><BR/>As for "data compression", I am using this terminology in a very general way. I am taking the reductionist viewpoint that physics emerges from something deeper and simpler than our low energy measurements currently see. This imposes no prior constraint on the form of the underlying algorithms that we use in our fundamental physics theories.<BR/><BR/>This includes the possibility that physics turns out to be an accidental and contingent outcome of a simple underlying process (e.g. one point in a "landscape" of possibilities). I know that it <I>may</I> be the case that physics turns out to be a mess "all the way to the bottom" (there may be no "bottom"), and that physics is merely the way it is for purely anthropic reasons; I am just betting that this isn't the way things are.<BR/><BR/>As for the type of "visualisation" that this needs, I am using this terminology in a very general way. I <I>don't</I> limit myself to the types of visualisation that are immediately accessible using the visual intuition that we develop by living in our world of sensory experience. The example in the blog posting is artificially simple. That's only where we <I>start</I>. I add to that the various layers of more abstract visualisation that we can develop to <I>extend</I> our powers of visualisation. This is an approach that I have gradually built up over many years, and which is greatly helped by using <I>Mathematica</I> as a symbiotic extension of my brain.<BR/><BR/>Am I purely calculational because I use "standard mathematics"? I use whatever tools I find are productive to use, and I haven't yet discovered anything that I can use to replace standard mathematics, although I <I>have</I> found ways of using various existing mathematical techniques in <I>new</I> ways. Inevitably, it will be true that there are yet-to-be discover (Platonic, yes!) mathematical techniques that will make my life much easier, but I am realistic enough to see that I am <I>not</I> going to be the one to discover these, because am a user not a discoverer of mathematical techniques. That's why I did a theoretical physics rather than a maths degree!Stephen Luttrellhttps://www.blogger.com/profile/05612770386489550404noreply@blogger.comtag:blogger.com,1999:blog-17232904.post-1328897406481148982008-02-03T17:47:00.000+00:002008-02-03T17:47:00.000+00:00Please don't assume that visualization of mathemat...Please don't assume that visualization of mathematics is itself, necessarily, the correct approach. Using the same logic as you employ here, it could be argued that a conventional framework implies a lack of visualization in the same context that you use here.<BR/>For example, in reply to a comment attached to your blog entry of Sep 12 you express data compression of the "standard model" as if it were a *necessary* way forward on the road to achieving a more complete encapsulization of physical law. However, physical law might well be centred on a system that processes symmetries based on a more fundamental understanding of what "symmetries" and "processing" as we understand it actually are. The behaviour of fundamental processes and forces will then reduce to a collection of symmetries that are "processed" according to a much more fundamental (and possibly very simple AND counter-intuitive)mechanism.<BR/>The way that works needs a profoundly deeper level of "visualization" and intuition than the abdication of the "calculation" based approach that you describe. In fact if I were to be unkind, I could accuse you of attempting a purely calculational approach by using a standard framework of mathematics at all.<BR/>However, I am not unkind; I'm just pointing out that there are far deeper levels of abstraction and convention than implied by your example.Anonymousnoreply@blogger.com