Sunday, December 09, 2012

"At root, Mathematics is about the manipulation of Qualities, not of Quantities."

Full Title:  "At its root, Mathematics is about the manipulation of Qualities, not of Quantities."




Dear Readers,



Below:  An excerpt from a recent dialogue on the F.E.D. mathematics of dialectics, in relation to mathematics as a whole, for your enjoyment --

Regards,

Miguel



Q:  You wrote:

1. Set Theory: if a, b, c, and d denote four distinct set elements, then:

A. the set {a, b} is not > the set {c, d}, and

B. the set {a, b} is not = the set {c, d}, and

C. the set {a, b} is not < the set {c, d}; therefore

D. the set {a, b} is unequal to, but not quantitatively unequal to the set {c, d}; therefore

E. the set {a, b} is qualitatively unequal to the set {c, d}.


[But] the operator '>' makes no sense at all in the context of sets ...

Besides, the conclusion '{a,b}' is qualitatively unequal to '{c,d}' is implied in the fact that they're 4 distinct elements ... All this stuff says is yeah both subsets contain the same amount of elements but they're not the same elements...



M.D.:   [Note also that the -- careless -- statement above commits the fallacy of identifying sets with their elements, whereas it is of the very essence of set theory that sets and elements are [qualitatively] distinct kinds of '[idea-]objects'. In set theory, even a "singleton" set is [qualitatively] unequal to its single-element content, if placed outside of that set: {a} is qualitatively-unequal-to a. The careless statement above also erroneously identifies '{a,b}' and '{c,d}' as "subsets", rather than just as sets[.

It was not the present writer who asserted that the purely quantitative relations of, e.g., the "Natural" Numbers, denoted '>' and '<', applied to set theory.

On the contrary, it was the interlocutor addressed by the post in question who asserted that, implicitly, by asserting, essentially, that '''all mathematics is [only] about the manipulation of quantities'''.

The post in question was designed, in effect, to provide that interlocutor with a <<reduction ad absurdum>> disproof of the [common] claim that '''all mathematics is [only] about the manipulation of quantities''', viz. --


Assumption: All of mathematics is only about the manipulation of quantities.

This means that all "mathematical objects" must be "quantities".

Set theory is [a part of] mathematics.

The 'mathematical objects' treated by/in "set theory" are named "sets".

The "trichotomy principle", the principle that holds for all purely-quantitative mathematical objects -- that holds for all "quantities" -- does not hold for sets, i.e., it is not true that for any distinct sets A and B, "A is > B, OR A = B, OR A < B."

Therefore, sets, the ''mathematical objects" of the mathematical theory named "set theory", are not "quantities".

Therefore the assumption/assertion that "All of mathematics is only about the manipulation of quantities." is a false assertion.


It is interesting, that, as noted, the distinct sets {a, b} and {c, d}, have the same number of elements -- the same "cardinality":  2 -- and yet the sets themselves are qualitatively different.

Set theory is often said to be used to deductively derive "quantity" and "quantities", e.g., to derive the "Natural" Numbers.

"Quantity" in such derivations begins as a QUALITY -- a "predicate" of sets -- called cardinality.

For example, taking all objects in the "universe of discourse", or "universal set", of such a derivation, the "Natural" Number, or "cardinal number", 3, might be identified with/as the set of all "triples" -- the set of all sets which contain exactly three distinct elements, three of the [qualitative] objects that are included in the "universe of discourse", or "universal set", of this derivation, as potential base elements of sets.

That "set of all distinct sets with exactly three distinct elements" would represent, as an "extension", the "intension", or "meaning" of "three-ness", the "quality" that all of the sets in this set of sets have in common.

Each of the sets that are the elements of this set would be qualitatively, not quantitatively, different, from every other set-element in this set.

Each of the elements/members of those set-elements would be qualitatively different, not quantitatively different, from every other member of those set-elements.

It thus appears that, per this set-theoretical account of Number, qualitative differences and qualities are primary in mathematics, while quantitative differences and quantities are secondary, merely derivative from the qualitative; merely derived.

Precisely the opposite of the view expressed by the interlocutor.

At its root, therefore, per its set-theoretical derivation, mathematics is about the manipulation of qualities, not quantities.

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