On why mathematics appear to be non-cosmic

Preface

I do fear that per­haps this post of mine (my fourth here) may cause a few nega­tive re­ac­tions. I do try to ap­proach this from a philo­soph­i­cal view­point, as befits my stud­ies. It goes with­out say­ing that I may be wrong, and would very much like to read your views and even more so any rea­sons that my own po­si­tion may be iden­ti­fied as un­ten­able. I can only as­sure you that to me it cur­rently seems that math­e­mat­ics are not cos­mic but an­thropic.


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There are so many quotes about math­e­mat­ics, from cel­e­brated math­e­mat­i­ci­ans, philoso­phers, even artists; some are witty yet too polem­i­cal to iden­tify as use­ful in a trea­tise that as­pires to dis­cuss whether math is merely an­thropic or cos­mic, and oth­ers are per­haps too fo­cused on the or­der it­self and thus come across a bit like the ex­pected fawn­ing of an ad­mirer to his or her muse.

Yet the ques­tion re­gard­ing math be­ing only a hu­man con­cept, or some­thing which is ac­tu­ally cos­mic, is an im­por­tant one, and it does de­serve hon­est ex­am­i­na­tion. I will try to pre­sent a few of my own thoughts on this sub­ject, hop­ing that they may be of use – even if their use is sim­ply to al­low for fruit­ful re­flec­tion and pos­si­ble dis­mis­sal.

It is ev­i­dent that math­e­mat­ics have value. It is also ev­i­dent that they al­low for tech­nolog­i­cal de­vel­op­ment. They do serve as a foun­da­tion for sci­en­tific or­ders that rest on ex­per­i­ment and thus are in­valuable. How­ever we should also con­sider what the pri­mary differ­ence be­tween math as an or­der and sci­en­tific or­ders (physics, chem­istry etc) eas­ily let’s us know about math it­self:

Pri­mar­ily math differs from sci­ence in that it se­cures that its re­sults are valid not from ex­per­i­ment, data and ob­ser­va­tion, but ax­iom-based proof. The use of proof in math is of­ten at­tributed to the first Greek math­e­mat­i­ci­ans, and speci­fi­cally to ei­ther the first Philoso­pher, Thales of Mile­tus, or his stu­dents, Anax­i­man­der and Pythago­ras. Eu­clid ar­gued that the first The­o­rem that math pre­sents is the one by Thales, which has to do with analo­gies be­tween parts of 2D forms (eg tri­an­gles) in­scribed in a cir­cle. The idea of a proof pro­ceed­ing from ax­ioms, of a The­o­rem, is fun­da­men­tal in math­e­mat­ics – and it also is a cru­cial differ­ence be­tween math and or­ders such as physics. Fields of sci­ence that have to do with ob­serv­ing (and in­ter­act­ing with) the ex­ter­nal world do sig­nifi­cantly differ from a field (math) which only re­quires re­flect­ing on ax­io­matic sys­tems.

Given the above is true, it does fol­low that a hu­man is far more con­nected to math than to any study of ex­ter­nal ob­jects: they are tied to math with­out even try­ing to be tied to it, given math ex­ists as a men­tal cre­ation and not one which re­quires the senses to in­ter­vene.

But what does “be­ing more con­nected” mean, in this con­text? Is math ac­tu­ally in­ter­twined with hu­man thought of all kinds? Ob­vi­ously we do not in­nately know about ba­sic “re­al­ities” of the ex­ter­nal world, such as weight and im­pact; the risk of a free-fall is some­thing that an in­fant has to first ac­cept as a re­al­ity with­out grasp­ing why it is so. On the con­trary we do, by ne­ces­sity, already have fun­da­men­tal aware­ness of the (ar­guably) most ba­sic no­tion in all of math­e­mat­ics: the no­tion of the monad.

The monad is the idea of “one”. That any­thing dis­tinct is a “one”, re­gard­less of whether we mean to in­clude it in a larger group or di­vide it to con­stituent parts: each of those larger groups are also “one”, and the same is true for any di­vi­sions. “One­ness”, there­fore, as the pre-so­crat­ics already ar­gued (and Plato ex­am­ined in hun­dreds of pages) is ar­guably one of the most char­ac­ter­is­tic hu­man no­tions, and a no­tion which is gen­er­ally in­escapable and ubiquitous. “One” is also the first digit and the me­ter of the set of nat­u­ral num­bers (1,2,3,4…), and this is be­cause the hu­man mind fun­da­men­tally iden­ti­fies differ­ences as dis­tinct, even when the differ­ence may be­come (in ad­vanced math) ex­tremely com­pli­cated and of pe­cu­liar types. Yet the hum­ble set of nat­u­ral num­bers also gives us an in­ter­est­ing se­quence when al­tered a bit: the so-called Fibonacci se­quence, which I think is a good ex­am­ple to use so as to show why I think that math are only hu­man and not cos­mic.

The Fibonacci se­quence pro­gresses in a very spe­cific way: each part is formed by adding the two pre­vi­ous parts. The se­quence be­gins with 1 (or 0 and 1), so the first parts of it are (0), 1, 1, 2, 3, 5, 8,13. The en­tire se­quence di­verges from both sides (al­ter­nat­ing be­tween the next part pre­sent­ing a nu­mer­i­cal differ­ence just smaller or just larger) to the golden ra­tio, and forms a pretty spiral form (wiki image: https://​​en.wikipe­dia.org/​​wiki/​​Fibonacci_num­ber#/​​me­dia/​​File:Fibonac­ciSpiral.svg). Yet for me it is of more in­ter­est that hu­mans do hap­pen to ob­serve a good ap­prox­i­ma­tion of this spe­cific, math­e­mat­i­cal spiral, on some ex­ter­nal ob­jects; namely the shells of a few small an­i­mals.

It is pretty clear that the shell of some ex­ter­nal be­ing is not it­self aware of math­e­mat­ics. One could ar­gue, of course, that “na­ture” it­self is filled with math­e­mat­ics, and thus in some way a few ex­ter­nal forms hap­pen to ap­prox­i­mate a spe­cific spiral, and the tie to the golden ra­tio etc is only to be ex­pected given na­ture (and by ex­ten­sion, per­haps, the Cos­mos it­self) is math­e­mat­i­cal. Cer­tainly this can ap­pear to provide an an­swer; or to be pre­cise it would at least pre­sent a cause for this ap­pear­ance of math­e­mat­ics and of a spe­cific spiral in the ex­ter­nal world. Is it re­ally a good an­swer, though? In other words, do we ob­serve the Fibonacci or golden ra­tio spiral ap­prox­i­ma­tion on the ex­ter­nal world be­cause the ex­ter­nal world it­self is tied to math, or do we do so be­cause we are tied to math in an even deeper way than we re­al­ize and could only pro­ject what we have in­side of our men­tal world onto any­thing ex­ter­nal?

My view is that hu­mans are so bound to math (re­gard­less of how knowl­edge­able one is in math­e­mat­ics) that we can­not but view the world math­e­mat­i­cally. Rock­ets are built, us­ing math, and by them we can even leave the or­bit of our planet – yet con­sider whether what al­lowed us to re­al­ize how to achieve so im­pres­sive a re­sult was not math alone, but math as a kind of very an­thropic cane or leg by which we slowly learned to move about:

In essence I do think that due to the hu­man species be­ing so ob­structed from de­vel­op­ing far more ad­vanced math­e­mat­ics (to put it an­other way: due to how difficult ad­vanc­ing math can be even for the best math­e­mat­i­ci­ans) we tend to not iden­tify that math it­self is not the cause of de­vel­op­ment, not the cause of move­ment and pro­gres­sion, but a leg—the only leg—we have to fa­mil­iarize our­selves with be­cause we as­pire to move on this plane. Imag­ine a dog which wanted to move from A to B, but couldn’t use its legs. At some point it man­ages to move one of them, and then enough so as to fi­nally get to B. It is un­doubt­edly a ma­jor achieve­ment for the dog. But the dog shouldn’t pro­ceed to claim that the dirt be­tween A and B is made of mov­ing legs – let alone that it is the case for the en­tire Cos­mos.

I only meant to briefly pre­sent my thoughts on this sub­ject, and wish to spec­ify (what very likely is already clear to more math­e­mat­i­cally-ori­ented read­ers of this post) that my per­sonal knowl­edge of math­e­mat­ics is quite ba­sic. I ap­proach the sub­ject from a philo­soph­i­cal and episte­molog­i­cal view­point, which is more fit­ting to my own Univer­sity stud­ies (Philos­o­phy).

by Kyr­i­akos Chalkopoulos (https://​​www.pa­treon.com/​​Kyr­i­akos)