On some quotes from P. Mancuso's "From Brouwer to Hilbert: The debate on the foundations of mathematics in the 1920s".

[Ross A. Finlayson]

On some quotes from P. Mancuso's
"From Brouwer to Hilbert: The
debate on the foundations of
mathematics in the 1920s".

This is a useful read for those
not too familiar with the background of mathematical foundations and
schools of thought on constructivism
and strong constructivism.

Brouwer's notion of the continuum
as primary fits well in some respects
with some modern contemporary
notions of the continuum and the
address and resolution of "fleeing
properties" about the reductio.

(The book contains commentary
and original translations of Brouwer
and Hilbert.)

Hilbert's outline of formalism's place
(and a complete formalism or
"Formalism") is of interest and note.

[Ross A. Finlayson]

Also Brouwer's notion of the conscientious
mathematician are rather refreshing, though
he couches them in a rather ineffective frame
for someone who claims not to be a Platonist.

This is where a conscientious mathematician can
be strongly platonist and that the numbers exist
absent our existence and etcetera.

[George Greene]

On Saturday, August 13, 2016 at 11:36:49 PM UTC-4, Ross A. Finlayson wrote:
> Also Brouwer's notion of the conscientious
> mathematician are rather refreshing, though
> he couches them in a rather ineffective frame
> for someone who claims not to be a Platonist.
>
> This is where a conscientious mathematician can
> be strongly platonist and that the numbers exist
> absent our existence and etcetera.

By modern standards these debates come off as plumb stupid.
Re Platonism, does the letter 'a' exist?
The fact that they never even thought to ask the question really
proves on some level that they were just clueless.

Why does it even matter whether the letter 'a' does or doesn't "exist"?!??

[Ross A. Finlayson]

I think we can agree that some of Brouwer's notions are
today somewhat passe (or backward), but there is a gainer
in Brouwer's notions of the formally constructive.

With 'a' as the first letter of the alphabet, everything that
a language of glyphs is has at least a letter in it. Here
we're talking about the commonalities besides the specificities.
This is where, for example, 1 is not just S(0) or 1/1 or 1.0,
it's all of those, here in terms of the language(s) of those
things.

Then, having the strongly platonist view, doesn't have to
be that everything is thusly discernible as an object of
pure mathematics, but rather, mathematics (logic) has all
that the discernibles or predicative can be, and even more.

It's almost as key to remind that there _is_ a foundational
debate. A lot of people gave up with Goedel. Then, where
set theory could have been closed (and somewhat humbly as
the regular and all) then Cohen broke it open about the
basic result of Cantor's set theory as well-foundedness
not necessarily corresponding to regularity. It's usual
that the "continuum hypothesis of cardinals" (CH) doesn't
see much meaning, but any post-modern treatment is going to
have to apologize and explain what is going on beyond
the regular set theory, in terms of a mathematical universe.

It's quite convenient then for mathematics and physics
that's it's a mathematical physical universe, and even
scientific, quite all so neatly in terms of concrete
resources that are mathematical objects, or numbers.

[Ross A. Finlayson]

"In my opinion, this is essentially what happens
in my investigations of the principles of mathematics.
The _a priori_ is here nothing more than and nothing
less than a fundamental mode of thought (Grundeinstellung),
which I also call the finite mode of thought: something
is already given to us in advance in our faculty of
representation: certain extra-logical concrete objects
that exists intuitively as an immediate experience before
all thought. ... This is the fundamental mode of thought
that I hold to be necessary for mathematics and for all
scientific thought, understanding, and communication, ..."

"In a recent philosophical lecture I find the sentence:

The nothing is the absolute negation of the allness of being.

...

It is precisely one of the most important tasks of proof theory
to present clearly the sense and admissibility of negation:
negation is a formal process by means of which, from a statement S,
another arises, which is bound to S be the axioms of negation
mentioned above (essentially the principle of contradiction and
_tertium non datur_). ..."



-- Hilbert, "The Grounding of Elementary Number Theory", 1931


Hilbert's quite the platonist then for what he demands the regular.

[burs...@gmail.com]

Hi fruitcake, you should really give a try:

"If one were writing merely the history
of one branch of mathematical logic or
founda- tional enterprise in isolation
from the rest, it would be difficult to
appreciate the combined magnitude of these
contributions of Kleene and Kreisel to
the history of logic in the last century.
No doubt we have not said all there is to
say about their contributions or that of
other figures such as Feferman or Friedman
whose cross-disciplinary work is better
known. None of these figures adhered to or
is associated with a specific foundational
standpoint. But our history has also served
to illustrate how their extension of results
and methods which grew out of these standpoints
served to guide the development of mathematical
logic in the second half of the last century –
albeit in indirect and occasionally
surprising directions."

The Prehistory of the Subsystems of Second-Order
Arithmetic Walter Dean* and Sean Walsh**
December 20, 2016
https://arxiv.org/pdf/1612.06219.pdf

Its not about the roaring 20s, but besides figures
such as Hilbert, Bernay, ... there are also newer
figures such as Friedman, Feferman, ...

it also contains a plea to not forget Kreisel and
Kleene. And since they worked all interdisciplinary
labeling them with some fixed standpoint,

is rather fruitless (pun intended).

[burs...@gmail.com]

Wanted to research interdisciplinarity,
but found this nice read:

Stairway to Heaven
The abstract method and levels of abstraction in mathematics
Jean-Pierre Marquis∗ D ́epartement de philosophie
Universit ́e de Montr ́eal
https://papyrus.bib.umontreal.ca/xmlui/bitstream/handle/1866/19804/Levelsofabs.pdf?sequence=1&isAllowed=y

Compare Hilbert with Grothendieck, both
were doing geometry. So what happened?

[burs...@gmail.com]

BTW: I have the feeling Grothendieck stuff
is even teached nowadays to math students.
Just check out some university curricula in
algebraic geometry.

[Ross A. Finlayson]

When you say "fixed point" that reflects on various
notions of comprehension besides "regular infinity"
(which doesn't so much per se admit a fixed point)
as about various systems as reflect usual notions of
supertasks, or completed infinity as for Zeno and
other usual notions of the success of infinite regress
or background of motion.

Kriesel's "modified" realizability as along the lines
of the "material" implication and its erasure of the
truth-value semantics in a land of "material" implication
vis-a-vis Kleene as following Lukasiewicz for the "undefined"
(as separate from for example simple methods in formal
automata) is rather unproductive indeed vis-a-vis the
20'th century notions of Quine, Popper, etcetera as then
usually those being philosophers albeit technical where
the logicians seem to fail to justify their "extra-logical"
initiative to satisfy the classical and beyond the simpy
regular as is so simple (and not complete).

Friedman and Feferman as basically exploring Goedel after
being handed the "larger" of the "cardinals", here as
well there is the plainly universal which is to be built
into the foundations not tacked over nor breaking the
first-order with the "second".

Seems a simpler matter of "infinity" after zero and
one.

Here the point for Hilbert is that exactly what he
demands in the regular he establishes as "_a priori_".

This then for the conscientious logician remains in
a general sense to put the universal and as the
intuitive (as it were, the deductive at the extremes
and beyond the "bounded constructivist") _before_ the
simple regular, equipping the theory with a foundation
instead of a fault.



You can keep your diminutive for yourself.



Hilbert requests a regular program after quite well
establishing for himself that there already is a
complete program.

So, maybe modern mathematics should start again with
that, simply finding neatly how to build all the
regular again but actually having a sound method
to appease the technical philosophers (who would
otherwise let the axiomatists go as "logical",
just not then for the overall, in the context,
"sound" or "constant, consistent, complete, and concrete".


The "modified" "material" is just poison in the well
(for truth-value semantics as having their values
immediately, not their invalidations as indeterminate).


So, the "second-order" as stratifying or tilling under
are quite opposite each other in a regular system where
it's only the one or the other of up or down (here per Hilbert).

Then, it might be enjoyed usual novel initiatives as re-write
just the initial critical bit, round and round a simple
mutual contradiction instead of "symmetry flex", but,
that's busy-work for logicians not satisfaction for
the conscientious constructivists formalist in foundations.

(Who just needs the one, with that neatly
there is just the one.)

[Ross A. Finlayson]

This was interesting.

[Ross Finlayson]

These day's it's the "2020s".

[Mostowski Collapse]

Ross A. Finlayson is a good example how meaning
gets replaced by attention. Just a chain of buzz words,
he even doesn't know himself what he is posting.

Interestingly this is the secret souce of ChatGPT:

GPT and BERT: A Comparison of Transformer Architectures
https://dev.to/meetkern/gpt-and-bert-a-comparison-of-transformer-architectures-2k46

Most cited paper:

Attention Is All You Need - Ashish Vaswani et al.
https://arxiv.org/abs/1706.03762

[Ross Finlayson]

A Brouwer concept of a"clump" continuum, and
a Finlayson concept of a "curve" continuum,
are not so different.

The Brouwer "clump swell knot"and the Finlayson "curve line ball",
have that Brouwer's grows inside and is deep while Finlayson's
is grown in the middle and is only two things: the volume of the ball
and the length of the line. Brouwer's is kind of "without form" while
Finlayson's rather "is form".

Yet, they're not so different, that Brouwer describes what is "a continuum
what its properties are that confound" while Finlayson a "a continuum
with its properties that are co-founded".

They're similar in that the points, or atoms of the element, are contiguous,
they're all proximal and proximate, they're all in the same neighborhood,
it's a trivial topology and everything is in it. (Or, Finlayson's is a space and
it's a trivial topology and everything is in it.)

The contiguity of the points is not an un-usual notion. Of course there's a
counter-point that in the field and those being the standard reals that
there's no least element greater than zero, these what would be "iota-values"
include what would be of "non-standard" real-value, in the real values.

The contiguity of points is not an un-usual notion, the continuity of points
as a contiguity of points is like Aristotle's continuum of a unit, and, also
it's like Spinoza's integer continuum of an infinity, here that Brouwer's
is a clump and Finlayson's is a unit, (Euclidean), hyper-ball, in 1-D a line
segment and a unit, like Aristotle's. Aristotle has these as last, in the
consideration of the outset of the thing, while Brouwer and Finlayson
have these first, and Spinoza can be considered first or last, whether
there's an infinity the integer or an infinity the integers, about whether
if there are infinite integers there are infinite integers.


There are all sorts notions "infinitesimals". Of course integral calculus
was called "infinitesimal analysis", for example about Price's late-1800's
textbook "Infinitesimal Analysis". There's Peano and some "standard
infinitesimals", Dodgson wrote on infinitesimals which are like a field,
there's Stolz and Veronese, Brouwer's not alone and it's not like there
was a Dark Ages of "infinity" between Aristotle and Cantor, where of
course Cantor was adamant there were _not_ infinitesimals, after
Robinsohn and hyper-reals there were "innocuous infinitesimals" that
are only a "conservative extension" of the field in terms of that in topology
that are like the neighbors all close up. There's though that by no means
are Robinsohn's infinitesimals the "usual" notion of infinitesimals, which
are "usually" much closer to the differential for the applied, of what is
called "real analysis" which has a nice formal development, of what is
also called "infinitesimal analysis" which may have the same formal development
and is beyond "finite differences". That is, there are usual sorts notions
of infinitesimals, that are about real analytical character. There's the
nil-potent and the nil-square, like a number greater than one or a number
less than one but greater than zero, that its power taken to infinity is
either infinity or zero.

Then, Finlayson called these "iota-values", and separated their addition
and multiplication, and gave them various analytical characters according
to the dimensions they exist in, the "polydimensional points" as of the
"pandimensional" points, where points are 1-sided "in" a line and 2-sided
"on" a line and 3, 4, or 5 sided, as minimal, "on", a plane, that points are
only "in" a line and "on" higher-order real surfaces or the continuous manifold.

It was after that, and for Vitali's measure theory why that "doubling spaces"
are a natural result of this model of the continuum, about measure theory
and doubling spaces and the measure 1.0 of points "in" a line and "Vitali 2.0"
the points in the doubled space "on" a line, that re-interprets the interpretation
of Vitali's analytical and geometrical point in terms of "doubling space" instead
of "non-measurable sets". This then is a natural setting for Vitali and Hausdorff
in terms of Banach and Banach-Tarski, as was discussed in "Banach-Tarski paradox",
that it's not a paradox but instead a function of the act abstracting an analytical
space from a geometrical space, in a particular example as of a signal over the
entire space. This makes geometry and an abstract space for analysis to be different.


So, Brouwer's and Finlayson's and Aristotle's continuums are not so different,
and, indeed, some have that they're constructive, and that it's not so much that
Brouwer was an "intuitionist" as that he was constructive in a different way so
there was named a sub-field of "constructivism" to "intuitionism". This is where
the usual non-constructive or as only demonstrated by contradiction settings of
the "uncountable" are non-constructive but for formal rigor must be built in the
constructive afterwards thus that as deemed "standard" they must be constructive,
that in a similar sense as the objective and subjective they are just as well, "intuitionist",
that being a just matter of perspective (and priority).

[Mostowski Collapse]

Ha Ha, Rossfinlayson thinks "reading" and "read
aloud" are the same? How moronic can one be?

Anyway, I just had a laugh at this read aloud. You
can try here, at t=652, and count the "eh", he has
really problems reading the text.

https://youtu.be/UkZGZ6FRpS0?t=652

If its not nigher pitch or lower pitch, he prolongs
words, or suddently speeds up, like instead "is
two", he says "is twooooooo".

LMAO!

[Mostowski Collapse]

But first of all the potato in the mouth
is very annoying. Even if the pornounciation
and the "eh" breaks would go, it would still

be a horrible read aloud, just unbearable.

[Ross Finlayson]

I was describing the usual orthornomal vector basis
and that R^3's is usually called e sub i or e sub 1,
so, generously, read out "e, e, e, e sub 1". Here the configuration
space being discussed is flux through a surface, with R^2 and
a time-like dimension.

It's a good stutter though if such things as the perceived
failings of other people are what activate your brain's pleasure
centers, where that's Shadenfreude and is a usual trait of
psychopaths and the sadistic. Surely it aggrieved me.

Please though, if I ever make an intonation that "it ends
like a question?" that something is terribly wrong, because
that is a grating-nails-the-chalkboard-the-teacher-doesn't-know,
and such "presenters" are "defective".

Also there's one good "uhhh....". Sega Dreamcast: "It's thinking."


Now that you've interjected after the wonderful preceding post, and
please feel certain that I read my own posts with certainty
and much as they are written as they are written to be read,
discuss Brouwer's concept of the continuum and see if it
makes sense or that otherwise perhaps it doesn't fit in your head.

[Mostowski Collapse]

Maybe its not a potato in Finalysons mouth,
rather he is sucking some cock?

[Ross Finlayson]

No, I hadn't thought so, though, you do have a dirty mouth.

It reminds me of Marines and "you have a pretty mouth" and
the old Texan "fight the dick or bite the dick", and, the old man
on the television segment "I'm so old the wind blows."

Here such psycho-sexual neurotics are ignored as palaver.

That's tragic though that you're still working on seventh-grade shame protocols.
Thus I wonder, when you wake up with a spoon and dream you were eating
chocolate: rather I don't wonder and don't want to know.

So, keep your psycho-sexual bullshit, it suits you.

Anyways such as your filth here are irrelevant to this great conceit
about Brouwer's continuum and Finlayson's continuum, and Aristotle's
continuum, and Spinoza's continuum, and continuum mechanics.

https://en.wikipedia.org/wiki/Continuum_mechanics

[Ross Finlayson]

It's like in South Park, one of the teachers is bemoaning to the other teacher,
that the kids are filthy and making gay fellatio references. So, the teacher
advises the other teacher, that as sort of a reverse psychology, when the
little shit goes "filthy act my dirty parts", that you go "present them", upping
the ante as it were, resulting that the little shit is stymied and results humbled.
Or, heh, "Mr. Garrison". Heartened with Garrison's advice, the teacher was able
to return to class, where the next day, Eric Cartman, who is one of the main
characters of this parodic adolescent cartoon, and we respect his authoritar
and the lotion is put in the basket, comes up with buffooning "filthy act my
dirty parts", to the sniggers of his little shit buds. So, the teacher sucks it up,
turns to Cartman, and goes "present them". It blows his mind and he matures,
if only that much.

Among teachers, "little shit" is a term occasionally bantered around the staff-room,
it refers to bad kids or delinquents or miscreants or products of bad upbringing
or future jailbirds. It's distinguished from good kids, who are polite to each other
and respectful to adults. One hopes the "child-parent-adult" emotional development
of the theory is developed as people gain emotional maturity, and one imagines
your mother will be very proud if she ever crawls out of the alley where she's doing
who-knows-what standing up against the dumpster behind the place you fell out.
The child-parent-adult emotional development is the idea that we start as children
and knowing our place as children dependent on our parents, then modeling them
learn the roles and adopt the roles of parent or providing for dependence, then
that as adults, we understand that everyone's just people and that "you're not my
father" goes both ways. You're not your brother's keeper, you're your brother's brother.

It reminds one of Richard "Tricky Dick" Nixon, whose uncensored moments exposed
that about every few words he uttered "dirty-parts-filthy-acter". It's among reasons
why he left office, because when it came out his filthy mouth totally betrayed his
public image, he had no credibility. And he left, and it was in bad taste.

Now, there's something you should know about humor, and not just the various
natural humors but what strikes the funny bone. It's not Seneca's extended treatise
on how to put the barbs in using rhetoric of the humorous sort that leaves the audience
amused instead of just plain angry, "How to make a joke (of someone and at their expense)".
It's that: there are only twelve jokes, or maybe thirteen, if you count funny. There are only
twelve jokes, the participants vary and in any language the outcomes are the same and
the humor is for the same reason. There are only twelve jokes, and, seven of these are
too dirty to tell. There are only twelve jokes, and everybody knew all of them by third grade.

Filed under humor....

Now, all this has to be with Mancuso's debate is that Burse is a litte shit,
and that humor is used to deflect his abasement.

Then there's that Mancuso's debate:

is again for foundations of continuity and that it's a wonderful book and very
well informed and much helps understand the significance and importance of
Brouwer in the outset of the constructive foundations as constructive foundations,
why then before Goedel and then before Cohen, who provided intuitionist outs
from what in hindsight would've been a failure of the "foundations", that the
important concepts of Aristotle and Spinoza and Duns Scotus of the integer continuum
and the clock continuum, these days after all of Peano and Dodgson and Veronese
and Stolz and for that matter "Internal Set Theory" of Nelson and ignoring Robinsohn's
"hyper-reals: infinitesimals in reals that don't have infinitesimals", and for Conway's
sur-real numbers, and Finlayson's new Aristotle's clock continuum, which fits with
Newton's and Leibniz' fluxions and Leibniz raw differential an infinitesimal, that
the continuum thus resulting individua filled infinitesimals is a new old thing again,
making to re-Vitali-ize measure theory, for modern definitions of continuity, not just Dedekind's,
but including Aristotle's/Finlayson's line-reals, field-reals like Eudoxus/Cauchy/Dedekind,
Finlayson's signal reals, and long-line reals as of the infinitarcalcul of Paul du Bois-Reymond.

[Mostowski Collapse]

Or learn some Mandarin, to have a better
feeling for patterns of stress and intonation.

How do you say" want" in Chinese?
https://www.youtube.com/watch?v=dS0UOylWKsA

LoL

[Ross Finlayson]

No, thanks, I don't care to stress my nose to differentiate meaning,
that while pu tong-hua or pu-tong hua or Mandarin is widely spoken,
there's moreso that mgong-gok Eng-ma rather mgong-gok Amer-ma
and that while wo mingzi zao Lo-say or wo minz-de dzao Lo-say, there's
also that ideograms are more a code-book than an interchange form for
the written, and while the "etymology" is small like man-house-tree-chicken
the basic forms of ideograms as of Chinese calligraphy it's not very direct
in terms of what roots result English and Germanic tongues after the
Romance languages after the Italic and Indic. Then, I don't mind that
Unicode has some 80,000 ideograms but pretty much only need Latin1,
including usual forms of phonetic encodings, like the schwa. I have the
ear for tone, but, it's reserved for emotive tone, and kept separate from
the words themselves, as it's ignored in written interchange, and confuses
the intent with the structure of the language.

I can speak much louder and deeper but that frightens some people,
and don't much care to yodel as it would be farcical. The diacritical and
accents I generally automatically translate to English or leave idiomatically,
as loan-words, and tone and cadence are instead employed to maintain cadence,
of long-winded utterances that are used to describe full and complete sentences.

Being a Mandarin is a different thing, vis-a-vis long-bearded gentlemen
with long fingernails or about being a Mandarin vis-a-vis wu wei.

So, keep a civil tongue in your head.

[Ross Finlayson]

Brouwer's Continuum is a great example of
a fruit-ful side-line conceptually,
that in the revisiting of questions in
the classics and the canon and the modern,
framing the alternatives, is like,
studying Newton's gravity and Einstein's gravity,
then having Fatio and LeSage, or Newton's light
and Einstein's light, and Fresnel and de Broglie.

[Ross Finlayson]

Brouwer's continuum reminds me of Peano's continuum, Stolz and Veronese,
Dodgson, Conway, Robinsohn, Cantor, or Weierstrass,
of course Duns Scotus and Spinoza, du Bois-Reymond, Bell's smooth,
Euclid's smooth and Poincare's rough, Archimedes and non-Archimedes,
Eudoxus through Dedekind, Aristotle, here it's mostly for Zeno,
Bishop and Cheng, Paris and Kirby, Nelson or Boucher,
then about particularly Vitali and Hausdorff, and the
doubling-spaces and halving-spaces,
among line-reals, field-reals, and signal-reals.

It's a continuum mechanics and continuum dynamics, ....