[FRIAM] Can empirical discoveries be mathematical?

Fri Sep 3 23:27:04 EDT 2021

Bravo!

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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 <desmith at 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, thompnickson2 at 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
> ThompNickSon2 at gmail.com
> https://wordpress.clarku.edu/nthompson/
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>
> *From:* Friam <friam-bounces at redfish.com> *On Behalf Of *Pieter Steenekamp
> *Sent:* Friday, September 3, 2021 12:48 PM
> *To:* The Friday Morning Applied Complexity Coffee Group <
> friam at 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, <thompnickson2 at 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
> ThompNickSon2 at gmail.com
> https://wordpress.clarku.edu/nthompson/
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