Bravo! --- Frank C. Wimberly 140 Calle Ojo Feliz, Santa Fe, NM 87505
505 670-9918 Santa Fe, NM On Fri, Sep 3, 2021, 9:09 PM David Eric Smith <desm...@santafe.edu> wrote: > Please allow me to try to make things worse, if I can. > > I worry that I may be partly responsible for the origin of this thread, in > my jabs at the analytical philosophers, who I think are responsible for…. > No, wait; I won’t start that again. > > In any case, I read Nick’s post as a good-faith effort to ask the question > productively, rather than scholastically or rabbinically. (Or > philosophically … No, no,…) > > Nick, do stay with the t-shirts. It is a better example for the question > you started with. When you go off on bachelors you are off in a narrow > corner of language and designation, which is a different question. > > You have made several discoveries, certainly empirical. I will use math > to say what they are, just because I have the language and it is shorter > that way. Mostly you have not yet “built” any math, and you probably can > only make a mathematical discovery once you are in some way operating > within a domain that is math. Here are some things I think you can say: > > 1. Whatever we mean by “space, as a place in which one can put things and > orient them” has a local coordinatization and geometry that is > characterized by the rotation group. Now you don’t yet know what “the > rotation group” is — to use that as a whole concept you would have to build > some math and show that it hangs together as a descriptive (meaning, > formal) system. But if you or anybody else builds that system, you can > claim the empirical discovery that whatever “space as a place to put > things” is, it has the rotation group as a symmetry of the orientations for > things. That discovery isn’t empty: lots of phenomena describable with > systematic language don’t have the rotation group as symmetries. The set > of all phylogenetic trees, or all strings of letters, don’t need any > description in terms of the rotation group. > > 2. If you had said a bit more, you might have observed that t-shirts have > orientations (in the topologist’s, rather than the direction-pointing, > sense). You can imagine putting the t-shirt into a mirror-image > configuration, since you can look at it in the mirror and clearly imagine > what such a t-shirt would look like, occupying space. But you can notice > that there is no way that by rotating or otherwise deforming it, that you > can produce the t-shirt in the mirror-image form. I would borderline give > you credit for a mathematical discovery here. You may not have the > language to express it, but you have the seeds of building such a language, > which is that there is a group of transformations that include the > reflections and the rotations, and the reflections are not reducible to the > rotations. > > 3. It could then be another empirical discovery to say that our physical > space-as-a-place-to-put-things is has that group as a symmetry group. > > 4. To be a bit more pedantic, you have discovered that t-shirts transform > under the SO(3) _representation_ of the rotation group. If you were not a > mathematician or a physicist, you would say “I had “the” group of > rotations; what is there to represent?” But a mathematician would tell you > that there are many representations of the rotation group, all having the > same algebra, yet different formal constructs, and a physicist would tell > you that electrons do not transform under the same representation as > t-shirts do. If you turn your t-shirt around in one full turn of a > pirouette (any axis is fine), it will be back the way it started. If you > do that for an electron, it will not be. You will have to do two full > turns of the pirouette for the electron to be back the way it was. Whether > it is a discovery or a construction by the mathematicians that there is > another concept (representation) beyond the concept (group) I won’t > belabor. I would say that mathematicians find that formal systems can be > built up in which groups and representations are different constructs, and > those formal systems can be made consistent, so whatever they instantiate > as “concepts” has a definite referent. But it is an empirical discovery > that electrons and t-shirts don’t transform the same way under rotations, > so however we package it in the math, we will need expressions that > correspond to at least two representations of the one group. (For > reference, the one for electrons is called SU(2).) > > So to belabor all this out is tiresome and requires a bunch of layers of > accumulation, but we can arrive at language habits that allow us to > organize our thoughts. > > Eric > > > > On Sep 4, 2021, at 2:21 AM, thompnicks...@gmail.com wrote: > > By discovery, I mean only happening on a regularity that was unexpected. > > I guess I didn’t need all the razzle-dazzle about the t-shirts. Let’s say > that I, being totally naïve of logic, announced to friam that I had made a > survey of all my never-married male friends and each and every one claimed > to be a bachelor. I offered to you-all, as an insight, that all unmarried > men are bachelors. I think I have made that “discovery” empirically; you > might have arrived at the same insight logically. Perhaps the empirical vs > mathematical thing is methodological. Of course, I now realize that > inorder to arrive at my empirical conclusion, I had to invoke the logical > form, induction: this man is un-married, this man is a batchelor, all > batchelors are unmarried. You might have arrived at the same conclusion > deductively (i.e., mathematically). > > Nick Thompson > thompnicks...@gmail.com > https://wordpress.clarku.edu/nthompson/ > <https://linkprotect.cudasvc.com/url?a=https%3a%2f%2fwordpress.clarku.edu%2fnthompson%2f&c=E,1,CxoV3-soQMap0aZ7-0ueqqGYjQKFXmEfLfybimqj7_3oKWdvM3OSq95UNkCQw22-kuoZ1z4snDbeGLXxf4kQ16gsp1RVHQERB_Lip55CaBk,&typo=1> > > *From:* Friam <friam-boun...@redfish.com> *On Behalf Of *Pieter Steenekamp > *Sent:* Friday, September 3, 2021 12:48 PM > *To:* The Friday Morning Applied Complexity Coffee Group < > friam@redfish.com> > *Subject:* Re: [FRIAM] Can empirical discoveries be mathematical? > > Nick, > > I quote from https://www.britannica.com/science/scientific-theory > "In attempting to explain objects and events, the scientist employs (1) > careful observation or experiments, (2) reports of regularities, and (3) > systematic explanatory schemes (theories). The statements of regularities, > if accurate, may be taken as empirical laws expressing continuing > relationships among the objects or characteristics observed." > > Based on this, I reckon, because you have reported the regularities, you > have discovered an empirical scientific law. Congratulations! > > Next is to systematically explain it, then you have a scientific theory! > > Maybe I did not answer your question? You asked if this is an empirical > discovery or a mathematical one. > > IMO you have done only the first part, the empirical discovery. This could > now be taken further and if you can prove it using formal mathematics, then > only can you claim you have made a mathematical discovery. So, it is (not > yet) a mathematical discovery. Sorry to blow your bubble. > > P > > On Fri, 3 Sept 2021 at 17:24, <thompnicks...@gmail.com> wrote: > > Colleagues, > > Years ago, my daughter, who knows I hate to shop, bought me a bunch of > plain T-shirts. The label’s on the shirts were printed, rather than > attached, and so have faded. Each morning, this leaves me with the problem > of decerning which is the front and which the back of the shirt, and even, > which the inside and which the out-. After years of fussing with these > shirts I decerned a pattern. Up/down, inside-in/inside-out, left/right, > front/back, crossed arms/uncrossed arms, you can’t do one transformation > without doing at least one other. > > Is this an empirical discovery or a mathematical one? > > I guess it boils down to whether “front/back” entails in its meaning > another transformation. Should we call empirical discoveries > “discoveries” and mathematical discoveries “revelations”? > > Nick > > Nick Thompson > thompnicks...@gmail.com > https://wordpress.clarku.edu/nthompson/ > <https://linkprotect.cudasvc.com/url?a=https%3a%2f%2fwordpress.clarku.edu%2fnthompson%2f&c=E,1,Gj0sAmr-90l8xsv85ZnwtVVEyp1HV_9DvDFSK5riP2nKQ9Iz50-jMjBz6azBtfMIKzbiDfEnPloTPHvtRjZACXZ1ENfnhj69C_aNxACYOJ7FvW8yRg,,&typo=1> > > - .... . -..-. . -. -.. -..-. .. ... -..-. .... . .-. . > FRIAM Applied Complexity Group listserv > Zoom Fridays 9:30a-12p Mtn GMT-6 bit.ly/virtualfriam > <https://linkprotect.cudasvc.com/url?a=http%3a%2f%2fbit.ly%2fvirtualfriam&c=E,1,QuUXl8-qUVPOPvjLDaTd9j4330SQWwM0CgH6_1Gvu7U81Neh4Cd15VNuWk8OfjpojIl6rh8SFzZ6ABIpqhQT0JIM_jtluw_U84kA1reLjuk,&typo=1> > un/subscribe http://redfish.com/mailman/listinfo/friam_redfish.com > <https://linkprotect.cudasvc.com/url?a=http%3a%2f%2fredfish.com%2fmailman%2flistinfo%2ffriam_redfish.com&c=E,1,9WjLP-Dka1BsXw_Ukd9fsgA4j8KxW2WRDFlAMtvspczJSCYqCJnFXtFICqSsveeIkTFaH-S8EcMWnQtwvqfXz4SvGQZjKYVuYvhpnCgPiwVJ7A,,&typo=1> > FRIAM-COMIC http://friam-comic.blogspot.com/ > <https://linkprotect.cudasvc.com/url?a=http%3a%2f%2ffriam-comic.blogspot.com%2f&c=E,1,dtNEwCKHVYp-6pGwZ8AbrYPavCxTLCZ37MvpdvhukdSA-6FctdE7VKT0L1oZNp8Wn6yVIoKzM4ZhQ59jzavRLE9ALL-z5K3364cG1L9UIQKjiA,,&typo=1> > archives: http://friam.471366.n2.nabble.com/ > > - .... . -..-. . -. -.. -..-. .. ... -..-. .... . .-. . > FRIAM Applied Complexity Group listserv > Zoom Fridays 9:30a-12p Mtn GMT-6 bit.ly/virtualfriam > un/subscribe > https://linkprotect.cudasvc.com/url?a=http%3a%2f%2fredfish.com%2fmailman%2flistinfo%2ffriam_redfish.com&c=E,1,kPgMy-0c7Ec3Ea3OgZZFwqZSD-zbd5KTxJp_2ae9nMXsrg-m80c8muIlJIUsVoiuUUgFUQgWDiIKjwKdcf8KJcdT_VJBlSrtACjR30X4k_Gysoxb_j1Zgcqkm1Tv&typo=1 > FRIAM-COMIC > https://linkprotect.cudasvc.com/url?a=http%3a%2f%2ffriam-comic.blogspot.com%2f&c=E,1,8jjdvF4xTIwoz0ArIAvZQWQRfv4urUfd8Z4p8aLiBTiXLn_Q9Lw5-LKMWAr2L9hPJ8h5alVG74OEXos0ztMH1CGUEmirtc4QKma4CMcsUF7imLMaav8kZA,,&typo=1 > archives: http://friam.471366.n2.nabble.com/ > > > - .... . -..-. . -. -.. -..-. .. ... -..-. .... . .-. . > FRIAM Applied Complexity Group listserv > Zoom Fridays 9:30a-12p Mtn GMT-6 bit.ly/virtualfriam > un/subscribe http://redfish.com/mailman/listinfo/friam_redfish.com > FRIAM-COMIC http://friam-comic.blogspot.com/ > archives: http://friam.471366.n2.nabble.com/ >
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