[FRIAM] Can empirical discoveries be mathematical?

Fri Sep 3 23:03:57 EDT 2021

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 <mailto:ThompNickSon2 at 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-bounces at redfish.com <mailto: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 <mailto:friam at redfish.com>>
> Subject: Re: [FRIAM] Can empirical discoveries be mathematical?
>  
> Nick,
> 
> I quote from https://www.britannica.com/science/scientific-theory <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 <mailto: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 <mailto:ThompNickSon2 at 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>
>>  
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