__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***, in relation to mathematics as a whole, for your enjoyment --*

**mathematics of**__dialectics__Regards,

Miguel

__: You wrote:__

**Q****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

*, but*

__un__equal to

**not**__itatively__

**quant**__equal to the set__

**un****{c, d}**; therefore

**E**. the set

**{a, b}**is

__itatively__

**qual**__equal to the set__

**un****{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***of '[idea-]objects'. In set theory, even a "singleton" set is [*

**distinct**__kinds__*]*

**qualitatively***equal to its single-element content, if placed outside of that set:*

__un__**{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 "

**N**atural" 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

*"*, the principle that holds for all

**trichotomy principle**"*mathematical objects -- that holds for all*

**purely**-**quantitative***"*-- does

**quantities**"__hold for__

**not****sets**, i.e., it is

__true that for any distinct sets__

**not****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

__assertion.__

**false**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

*.*

__qua__litatively differentSet theory is often said to be used to deductively derive

*"*and

**quantity**"*"*, e.g., to derive the "

**quantities**"**N**atural" Numbers.

*"*in such derivations begins as a

**Quantity**"*-- a*

**QUALITY***"*of sets -- called

**predicate**"*.*

**cardinality**For example, taking all objects in the "universe of discourse", or "universal set", of such a derivation, the "

**N**atural" 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

*"*

__tension", the__

**ex***"*tension", or "meaning" of "three-ness", the "quality" that all of the sets in this set of sets have in common.

**in**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,

*and*

__qual__itative differences*are primary in mathematics, while quantitative differences and quantities are secondary, merely derivative from the qualitative; merely derived.*

**qualities**Precisely the opposite of the view expressed by the interlocutor.

At its root, therefore, per its set-theoretical derivation,

*, not quantities.*

**mathematics is about the manipulation of**__qualities__
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