![]() I find getting back to dy and dx and dt or curly versions not being "mere" symbols but infinitesimal hyperreals as symbolised is a reassuring bridge between Physics and Math. And on this the body of knowledge becomes a compendium that is then useful. I forgot, we also have a built up body of known facts that constrain model building in subtle ways through the community and through the sense of intuition of the individual.Ībove, we are seeing, go to the Hyperreals and voila, things start to come out of algebraic manipulations that tie to sequences, series, limits, thus rates and accumulations of change. So, I am finding that axiomatic constructions of systems using framing propositions is going to be helpful. And if just part of the contingencies of some PW that is close enough to our own, they can speak again to us. So, we see abstract logic model worlds that we may construct that then lead to key entities that if necessary are framework to any possible world. Thus, bare distinct identity and coherence focussed on quantities will not cause things by force of self-action but by being constraints on being they lay out what can or must be or cannot be or happens not to be. As above, certain quantities and structures tie into the identity of any distinct possible world, opening up the logical structures and patterns that flow from it. My sense is, logic of being, possible worlds and necessary, world framework beings are key pointers as I discussed months back. My thought is, that Wigner posed a really really powerful challenge in asking why is Math so effective. We have an interesting discussion developing here. KF kairosfocus OctoOctober 10 Oct 26 26 2019 12:29 PM 12 12 29 PM PST Copy Comment Link JB & PaV, ![]() Then we can go to space variables and more abstract cases. Thence, fundamental theorem of the Calculus and mutually inverse operations. How do you define a slope of a curve? How do you get area under a curve?Ĭhord-tangent and parallel strips, thus infinitesimals. big applications all over the place, even in marketing, economics, dynamics of movemennnnts etc. But what of varying flow - I use a more or less Gaussian impulse and its sigmoid growth curve. Steady flow, linear plots, slopes and areas fall into place. Think, spiral spider's web.įor this, water running into a cylindrical glass, so rates and accumulations come all together. As a spiral curriculum thinker I tend to go with an opening shot key case study that a learning activities loop hits on lines that run out to a cluster of key ideas as anchor points. Change, flow, rates point to TIME as a logical first independent variable. Accordingly we have motivating context galore. I found that rates and accumulations of change are crucial and widespread. In fact, using these types of numbers, Calculus becomes pretty much identical with Algebra. In any case, from a practical side, infinitesimals make calculus, limits, and other sorts of mathematical ideas a LOT simpler to work with. However, in the 1960s, infinitesimals were proved to be equally mathematically rigorous as other standard mathematical entities. ![]() This move was largely philosophically motivated, with Hilbert and others trying to naturalize mathematics. Infinitesimals were essentially banned from mathematics in the 1800s because it was said that they were inconsistent and non-rigorous (this is why calculus switched from infinitesimals to limits). First of all, I think that infinities and infinitesimals are somewhat of the equivalent of Intelligent Design for mathematics. I find the hyperreals interesting for a number of reasons.
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