Helmut, List
In general algebra, we define particular types of structure formed by a set
with one or more "internal" laws of composition possessing certain
properties. I voluntarily leave aside the external laws with operator
domains on the set. The laws themselves are ways of associating any two
elements noted with conventional signs to obtain a third. Thus the sign "+"
has absolutely nothing to do with addition as you practice it in the real
world. The simplest structure consisting of a set of elements (without real
existence) and an internal law of composition that we can write as we want
(with "+" if we want) is called a magma... this abstract structure can be
implemented in a multitude of concrete sets in which we have observed that
the elements associate themselves two by two to form a third belonging to
the same set. For example, the set of natural integers (with or without the
zero) is a "magma." But it is more than that. It is also a semigroup as
soon as we establish that its law is associative, semigroup being a name
conventionally chosen to designate this more complex advanced type of
abstract structure. We can also say "associative magma." The natural
integers are still part of it. Now the set {1,2,3} consisting of the
numbers 1,2 and 3 alone is not a magma for the usual addition. On the other
hand, it has a structure of another kind, listed in the mathematical
repository, under the name of "total order structure" (so a fortiori it is
also a Poset = Partially Ordered Set) when we provide it with the natural
order relation. But if we call the three elements X, Y, and Z and if we
provide their set with an order relation defined in such a way as to "copy"
the previous relations by substituting X to 3, Y to 2, and Z to,1 we will
obtain an abstract structure which can be implemented in many other 3
elements sets, like for example (taken at random ;-) ) Peirce's categories
considered with their interdependence relations (involvement). This is why
your example of Aliens has no sense in algebra because the abstract
elements that we compose according to your addition is an "algebraic
unicorn." Indeed, the starting elements are fixed, unalterable, and all the
more unalterable because they have no real existence, even if the letters
that designate them are on Earth. If the Aliens want to call this way of
altering objects on their planet "+," they can, but they will not practice
the algebra that is practiced on Earth.
Best regards,
Robert MartyHonorary Professor; Ph.D. Mathematics; Ph.D. Philosophy fr.wikipedia.org/wiki/Robert_Marty *https://martyrobert.academia.edu/ <https://martyrobert.academia.edu/>* Le jeu. 12 août 2021 à 16:45, Helmut Raulien <h.raul...@gmx.de> a écrit : > Robert, Edwina, List > > The skeleton metaphor for a poset makes sense to me. Is it also a good > metaphor for mathematics being the skeleton of all other sciences? > I earlier wrote, that mathematics is based on axioms, and axioms are not > hypothetical, but inducted. Edwina asked what I mean by axioms. I admit, > that I donot understand the axiom- (inducted relation to the real world-) > character of these things. For example, "a + b = b + a" is said to be an > axiom. But to me it only seems to be a tautology of the definition of the > plus-operator (summation). Summation is defined as a not temporal, and not > sequential, only spatial connection. So where is the axiom? If axioms > really exist, these are connections to the real world, which are not > hypoteses (abductions), but experiences (inductions). This is how "axiom" > is defined, I think. But I until now dont see the so-called axioms as > axioms. > > Best, > Helmut > > > 12. August 2021 um 10:39 Uhr > "robert marty" <robert.mart...@gmail.com> > wrote: > > > Helmut, List > > > The case of Peirce's semiotics is different from that of the empirical > sciences...it does not require induction to be verified and in this sense > it can be said to be "robust" a priori. Indeed, the mathematical modeling > by an abstract structure of Poset isomorphic to the organization of > universal categories by involvements is purely qualitative and does not > require any experimental verification. CP 3.559 has been quoted a lot but > each one has drawn partial arguments from it. The part that I have > personally underlined ( see Podium, p.4) does not suffer any dispute and > leads us on the way to understanding this false debate between "formalists" > and "empiricists" reformulated in a kind of will of power of mathematics on > phaneroscopy. Here is this part: "*The skeletonization or > diagrammatization of the problem serves more purposes than one; but its > principal purpose is to strip the significant relations of all disguise*". The > Poset is nothing else than "*the skeleton-set*" of all phanerons, a > tri-relation of elements a priori in all that is present to any mind. But > there are many other things in the phaneron! The metaphors of the X-ray of > the skeleton of a vertebrate allow us to illustrate this point > (phaneroscopy vs. radioscopy): radioscopy reveals the skeleton by making it > appear within an image. The scanner, by taking over with the help of > computers (tomodensidometry) can show it in three dimensions and from all > angles. All the rest is obliterated and is part of the "disguise", i.e. all > the rest in which it is immersed; the technique of X-rays allows the > extraction of the skeleton (Wilhelm Röntgen, the first Nobel Prize of > physics gave them the usual name of the unknown in mathematics, X! ). > Continuing the metaphor, my "trichotomic machine" is a kind of scanner > perfected to radiograph the signs based on the preliminary phaneroscopy of > each of the elements of the sign but respecting the determinations of their > tri-relation. > > Regards, > > Robert Marty > > > Honorary Professor ; PhD Mathematics ; PhD Philosophy > fr.wikipedia.org/wiki/Robert_Marty > *https://martyrobert.academia.edu/ <https://martyrobert.academia.edu/>* > > > Le jeu. 12 août 2021 à 00:08, Helmut Raulien <h.raul...@gmx.de> a écrit : > >> Edwina, List >> >> I dont think that De Tienne "tells us that it is essentially detached and >> isolate from the Real World to be almost irrelevant". After all, >> mathematics is based on axioms, which come from the real world. These are >> premisses. What I donot understand is, why these are called "purely >> hypothetical" (in this thread). A hypothesis is a result of an abduction, >> but axioms are results of induction. >> >> Now, how mathematics further deals with these premisses, is said to be >> deductively. I think so too. But you, with Peirce, call it "reasoning with >> specially constructed schemata". This to me seems completely different from >> pure deduction. I donot undertand what is meant by it. Can you give an >> example for such a constructed schema? >> >> Best, >> Helmut >> >> >> 11. August 2021 um 19:54 Uhr >> "Edwina Taborsky" <tabor...@primus.ca> >> wrote: >> >> >> Bernard, JAS, Gary F, Robert, list: >> >> The problem I have with De Tienne's outline of mathematics is the intense >> focus he gives to its essential irrelevance to we who live in the real >> world. I don't think it can be ascribed to his sense of humour. He repeats >> it often enough that we must consider that he takes this view very >> seriously. >> >> And - I don't agree with JAS's view that this focus is merely to >> differentiate 'pure' from 'applied' mathematics. We have to instead, ask >> WHY Peirce emphasized the role of pure mathematics in his SCIENCES. Surely >> it has a role in our scientific exploration and analysis of our Real World? >> Otherwise - why do it? >> >> Yes, mathematics "deals exclusively with hypothetical states of things >> and asserts no matter of fact whatever, and further, that it is thus alone >> that the necessity of its conclusions explained" 4.232. And Peirce warns >> against what we can consider as the'induction' process as the sole sense of >> information, with his comment "to assert that any source of information >> that is restricted to actual facts could afford us a necessary knowledge, >> that is, knowledge relating to a whole general range of possibility, would >> be a flat contradiction in terms" 4.232. >> >> And he moves, not into applied mathematics but into phenomenology, for >> 'Thinking in general terms is not enough. It is necessary that something >> should be DONE. In geometry, subsidiary lines are drawn. In algebra >> permissible transformations are made. Thereupon the faculty of observation >> is called into play...Theorematic reasoning invariably depends upon >> experimentation with individual schemata...theorematic or mathematical >> reasoning proper, is reasoning with specially constructed schemata" 4.233. >> >> And "phenomenology, which does not depend upon any other positive science, >> nevertheless must, if it is to be properly grounded, be made to depend upon >> the Conditional or Hypothetical Science of Pure Mathematics, whose only >> aim is to discover not how things actually are, but how they might be >> supposed to be, if not in our universe, then in some other. A phenomenology >> which does not reckon with pure mathematics....will be the same pitiful >> club-footed affair that Hegel produced" 5.40. >> >> My understanding then, is that pure mathematics provides hypothetical >> models of reality - and then, 'something is DONE [Peirce's emphasis] with >> these hypothetical models....observations, experimentation with individual >> schemata. This is NOT applied mathematics! It is: to repeat: Thereupon the >> faculty of observation is called into play...Theorematic reasoning >> invariably depends upon experimentation with individual >> schemata...theorematic or mathematical reasoning proper, is reasoning with >> specially constructed schemata" 4.233. >> >> Thus, from my reading of Peirce, I come up with a different understanding >> of pure mathematics from that of De Tienne, who tells us that it is >> essentially detached and isolate from the Real World to be almost >> irrelevant. >> >> Edwina >> >> >> >> >> >> >> >> >> >> On Wed 11/08/21 10:45 AM , Jon Alan Schmidt jonalanschm...@gmail.com >> sent: >> >> Bernard, List: >> >> I agree with Gary F.'s reply just now, but already drafted this one so I >> am going ahead and posting it. >> >> >> BM: This is clearly a blunder since if the world stopped existing, there >> would no more exist mathematicians at all, neither pure nor applied. >> >> >> I would call it hyperbole rather than a blunder. The point is that for >> Peirce, pure mathematics does not concern itself with whether or not its >> hypotheses correspond to anything that exists. >> >> >> BM: Writing such a definitive judgment is just ignoring the every day >> work of mathematicians who pass their time in diverses experiments with >> forms, abstracts figures, models, constructs, etc., not to speak of the >> value of their underlying hypotheses. >> >> >> As I have noted before, Peirce states very clearly that unlike all the >> other sciences, pure mathematics is not a positive science. The >> "experiments" conducted by pure mathematicians take place entirely in >> the imagination, although often aided by concrete tokens of the relevant >> diagram types. >> >> >> BM: And since it will be repeated in the following slide, it has an >> intended purpose: to show that pure mathematics are internally coherent >> wild dreams cut off [from] the world. >> >> >> Indeed, that is consistent with Peirce's explicit definitions of pure >> mathematics >> as reflected in the numerous quotations that I have previously provided, >> although I would substitute "ideal hypotheses" for "wild dreams." >> >> Regards, >> >> Jon Alan Schmidt - Olathe, Kansas, USA >> Structural Engineer, Synechist Philosopher, Lutheran Christian >> www.LinkedIn.com/in/JonAlanSchmidt - twitter.com/JonAlanSchmidt >> On Wed, Aug 11, 2021 at 9:40 AM <g...@gnusystems.ca> wrote: >> >>> Bernard, list, >>> >>> Yes, you can regard De Tienne’s statement about mathematicians in a >>> non-existing world as a logical blunder; I regard it as a manifestation of >>> his peculiar sense of humor. >>> >>> As for the experience of mathematicians doing pure mathematics, you can >>> indeed call it “experience,” but only in a peculiar sense which is contrary >>> to Peirce’s regular usage. Usually in Peirce, the distinction between the >>> internal and external worlds corresponds directly to the difference between >>> a “world of imagination” and “the actual world.” The idea of externality is >>> virtually identical with the idea of Secondness and is closely related to >>> the metaphysical idea of reality. Peirce usually refers to “experience” >>> as something forced upon us, indicating that Secondness is essential to >>> it. In these Peircean terms, the “everyday work” of mathematicians, insofar >>> is it is purely hypothetical, takes place in an internal world, a realm >>> of “degenerate Secondness” (EP1:280, W6:211). >>> >>> As JAS has been reminding us, the context of De Tienne’s talk/slideshow >>> involves a focus on pure mathematics and a corresponding neglect of >>> mathematical applications. This is one reason why he (and Peirce) do >>> not refer to pure mathematics as “experiential” in the sense that >>> phaneroscopy is. >>> >>> Gary f. >>> >>> From: peirce-l-requ...@list.iupui.edu <peirce-l-requ...@list.iupui.edu> On >>> Behalf Of Bernard Morand >>> Sent: 11-Aug-21 09:18 >>> To: peirce-l@list.iupui.edu >>> Subject: Re: [PEIRCE-L] André De Tienne: Slow Read slide 23 >>> >>> Gary f. , list >>> >>> De Tienne slide 23 starts with: "BECAUSE mathematics, in principle, is >>> not concerned with anything but itself. The world could stop existing, but >>> to pure mathematicians that would at most be an inconvenience." >>> >>> This is clearly a blunder since if the world stopped existing, there >>> would no more exist mathematicians at all, neither pure nor applied. >>> >>> It is repeated in slide 24 that you published today: "The significance >>> and truth-value of such constructs [those of mathematicians] depends only >>> on their internal inferential coherence, not on the world of experience >>> ." >>> >>> Writing such a definitive judgment is just ignoring the every day work >>> of mathematicians who pass their time in diverses experiments with >>> forms, abstracts figures, models, constructs, etc., not to speak of the >>> value of their underlying hypotheses. >>> >>> The slide 23 blunder that you minimize as "a choice of language" is >>> certainly a good rhetorical trick to get the laughs on one's side. But this >>> is not a valid scientific argument. And since it will be repeated in the >>> following slide, it has an intended purpose: to show that pure mathematics >>> are internally coherent wild dreams cut off the world. >>> >>> In fact I think that the human ancestors of mathematics were those >>> prehistoric people who managed to figure out on the walls of their caves >>> the drawings of savage animals. >>> >>> I wish that at the end of this slow reading you will undertake the >>> phaneroscopic observations of mathematicians at work, without any prejudice >>> as Peirce suggested it. >>> >>> Bernard Morand >>> >> >> _ _ _ _ _ _ _ _ _ _ ► PEIRCE-L subscribers: Click on "Reply List" or >> "Reply All" to REPLY ON PEIRCE-L to this message. PEIRCE-L posts should go >> to peirce-L@list.iupui.edu . ► To UNSUBSCRIBE, send a message NOT to >> PEIRCE-L but to l...@list.iupui.edu with UNSUBSCRIBE PEIRCE-L in the >> SUBJECT LINE of the message and nothing in the body. More at >> https://list.iupui.edu/sympa/help/user-signoff.html . ► PEIRCE-L is >> owned by THE PEIRCE GROUP; moderated by Gary Richmond; and co-managed by >> him and Ben Udell. >> _ _ _ _ _ _ _ _ _ _ >> ► PEIRCE-L subscribers: Click on "Reply List" or "Reply All" to REPLY ON >> PEIRCE-L to this message. PEIRCE-L posts should go to >> peirce-L@list.iupui.edu . >> ► To UNSUBSCRIBE, send a message NOT to PEIRCE-L but to >> l...@list.iupui.edu with UNSUBSCRIBE PEIRCE-L in the SUBJECT LINE of the >> message and nothing in the body. More at >> https://list.iupui.edu/sympa/help/user-signoff.html . >> ► PEIRCE-L is owned by THE PEIRCE GROUP; moderated by Gary Richmond; >> and co-managed by him and Ben Udell. > >
_ _ _ _ _ _ _ _ _ _ ► PEIRCE-L subscribers: Click on "Reply List" or "Reply All" to REPLY ON PEIRCE-L to this message. PEIRCE-L posts should go to peirce-L@list.iupui.edu . ► To UNSUBSCRIBE, send a message NOT to PEIRCE-L but to l...@list.iupui.edu with UNSUBSCRIBE PEIRCE-L in the SUBJECT LINE of the message and nothing in the body. More at https://list.iupui.edu/sympa/help/user-signoff.html . ► PEIRCE-L is owned by THE PEIRCE GROUP; moderated by Gary Richmond; and co-managed by him and Ben Udell.
