
Has philosophy ever clarified mathematics? - Dawny33
http://mathoverflow.net/q/182215/85415
======
philofcompguy
As you may know, many mathematicians turned philosophers while trying to do
work on the foundations of mathematics. It seems like logic was the gateway
discipline. What we now know as the analytic turn in philosophy of early 20th
century came from such as lineage before it devolved into philosophy of
language. Frege, Russel, Whitehead, and Wittgenstein are well known in this
line of thought. However, the history of philosophy and mathematics goes way
back to pre-socratic philosophers like Pythagoras and centuries later Aquinas
and then Descartes. The question posed is rather strange given that
mathematical development has often been formed by philosophical thought. I
guess by clarifying the author means providing solutions (since he/she
mentions that "mathematical insight" is nowhere to be found in the
literature)? Since philosophy is not the same as doing mathematics the only
kinds of clarification that philosophy will provide is in terms of
distinctions, definitions and criteria: what is a proof, etc. This is because
'philosophy of x' is always meta discipline.

------
platz
Carlo Rovelli: "Why Physics needs Philosophy"

[https://www.youtube.com/watch?v=IJ0uPkG-
pr4](https://www.youtube.com/watch?v=IJ0uPkG-pr4)

A few notes from the above:

* the beginning of astronomy == plato's school

* the scientific method as falsification == Popper

* quantum theory / relativity / Heisenberg == positivism (If I don't see it (e.g. electron orbitals) it doesn't exist) (* e.g. complementarity)

* Einstein claimed that his reading of Schopenhauer was crucial to thinking about time, space, etc...

in essence, you are doing philosophy when you're re-evaluating your
methodology and using a evolving reflective feedback loops to change your
thinking.

~~~
euyyn
> quantum theory / relativity / Heisenberg == positivism (If I don't see it
> (e.g. electron orbitals) it doesn't exist)

Wow, that is such a non-sequitur that it casts doubt on the whole thing.

~~~
mannykannot
Furthermore, I believe there was quite a lot of physics done before Popper
explained how to do it. He said many valid and insightful things, but it was
not necessary for physics for him to say them.

~~~
platz
Popper didn't explain "how to do physics", nor did he invalidate previous
physics, nor did he "invent the scientific method". But would you not agree
that he influenced how scientists interpreted the scientific method? This is
the point I'm making.. please don't reduce it into a blanket idea such as
"phsyics could not have been done before Popper" which isn't anywhere close to
what I said.

~~~
mannykannot
I think it is close to what you said, though apparently not what you meant.

What this discussion is sorely lacking is actual examples, so let me offer one
up: reproducibility has lately become an issue in several branches of science,
and maybe it would not have done so without Popper's insights.

------
voidhorse
Frege, Russell, Whitehead, Pierce--of course!

It doesn't help that many philosophers of mathematics are, for obvious
reasons, either also logicians or mathematicians, so demarcating between
advancements in philosophy of mathematics that clarify mathematics and
advancements in mathematics that clarify mathematics can be a bit of a fool's
errand.

Whatever the case, I dislike it when folks from the sciences or mathematics
try to discredit or dismiss philosophy--funnier still, and luckily not as bad,
is when they question the _value_ of philosophy without realizing that
question is in and of itself a highly philosophical question!

Philosophy has been around for a long time and isn't going anywhere in the
perceivable future (though I suppose it depends on what metaphysics of time
you subscribe to :) ).

------
woodrowbarlow
On the flip side, here's an example of mathematicians clarifying philosophy.

[https://en.m.wikipedia.org/wiki/Mike_Alder#Newton.27s_flamin...](https://en.m.wikipedia.org/wiki/Mike_Alder#Newton.27s_flaming_laser_sword)

~~~
naasking
> On the flip side, here's an example of mathematicians clarifying philosophy.

But only by doing philosophy!

------
decasteve
Bill Lawvere was heavily influenced by philosophy in his [revolutionary]
contributions to the development of Category Theory [1][2].

[1]
[https://ncatlab.org/nlab/show/William%20Lawvere#RelationToPh...](https://ncatlab.org/nlab/show/William%20Lawvere#RelationToPhilosophy)

[2] [http://philosophy.stackexchange.com/questions/9768/have-
prof...](http://philosophy.stackexchange.com/questions/9768/have-professional-
philosophers-contributed-to-other-fields-in-the-last-20-years/9814#9814)

------
scythe
In some sense Grothendieck's investigations could be considered
"philosophical"; in the early 20th century algebraic geometers studied objects
called "varieties" and Grothendieck's coup resulted from asking the question
"what is the general class of object with which we can do algebraic
geometry?". Today scheme theory is entirely mathematical, but a scheme had to
be conceived as a philosophical concept first. The link also mentions Turing's
elucidation of the Turing machine as a similar process.

There is also the case of Frank Ramsey and Piero Sraffa, who were the only
close friends of Ludwig Wittgenstein, and who went on to make major
epistemological contributions to economics (and Ramsey was a philosopher in
his own right): Ramsey was the first person to really clarify the concept of a
subjective probability, and Sraffa was central in the capital aggregation
controversy.

~~~
pron
I am not familiar with Grothendieck's work and with algebraic geometry in
general, but Turing's work was rather unique, and very much philosophical in
nature[0].

I am basing the following on Turing's 1936 paper, _On Computable Numbers_ [1],
on Juliet Floyd's chapter on Turing's mathematical philosophy[2], and on Robin
Gandy's 1988 paper, _The Confluence of Ideas in 1936_ [3].

The generalization of mathematical concepts has always been an important part
of mathematics. In fact, Gandy writes that this was precisely the concern with
Church's claim that the lambda calculus can express all algorithms ("a vague
and intuitive notion", in Church's words). It was thought that perhaps a
genius could some day generalize the notion further to include lambda-
definability as well as things that fall outside it. But then Turing appeared
with arguments of a very different kind. For one, his construction of an
automatic machine, while precise, was completely arbitrary. It was clear to
everyone from his treatment that while his math concerns a specific formalism
(what's known as the Turing machine ever since Church named it so), it really
covers _all_ formalisms. For another, his proof of undecidability is very
different from the one normally presented as the proof for the undecidability
of the halting problem (which is really due to Martin Davis). Turing's proof,
Floyd notes, makes no use at all of _any_ logical construction, and does not
rely even on logical contradiction or the law of excluded middle (which are
expected to hold in many "reasonable" formalisms) but on more basic
philosophical principles that must hold in any and all formalisms.

Gödel called Turing's achievement a "miracle". In 1946 he said (quoted in
Gandy's paper):

 _...with this concept one has for the first time succeeded in giving an
absolute definition of an interesting epistemological notion, i.e., one not
depending on the formalism chosen. In all other cases treated previously...
one has been able to define them only relative to a given language, and for
each individual language it is clear that the one thus obtained is not the one
looked for. For the concept of computability, however,... the situation is
different. By a kind of miracle it is not necessary to distinguish orders, and
the diagonal procedure does not lead outside the defined notion..._

[0]: Turing was a mathematical philosopher among other things, and is known
for his discussions with Wittgenstein, published in _Wittgenstein 's Lectures
on the Foundations of Mathematics_.

[1]:
[https://www.cs.virginia.edu/~robins/Turing_Paper_1936.pdf](https://www.cs.virginia.edu/~robins/Turing_Paper_1936.pdf)

[2]:
[https://mdetlefsen.nd.edu/assets/201037/jf.turing.pdf](https://mdetlefsen.nd.edu/assets/201037/jf.turing.pdf)
to appear in _Philosophical Explorations of the Legacy of Alan Turing_

[3]:
[http://dl.acm.org/citation.cfm?id=213993](http://dl.acm.org/citation.cfm?id=213993)
published in _The Universal Turing Machine A Half Century Survey_ , 1995

------
grandalf
Isn't the crux of this back and forth proofs that rely upon the axiom of
choice vs those that do not?

It is my understanding that philosophers have added a lot to our understanding
of how the axiom of choice impacts logical reasoning about things which matter
to humanity in concrete ways.

~~~
gpawl
What proofs?

Axiom of Choice cannot matter to humanity in concrete ways (except as an
artistic activity), because the Axiom of Choice only applies to uncountably-
large sets, which have no realizable physical meaning in the Universe.

~~~
raverbashing
It's an interesting question

However while there are no physical realizations of infinite sets, they're
used all the time in math

So I guess mathematical entities do not need to be connected to physical
elements

~~~
evincarofautumn
Of course. It’s not clear that the “real” or even “natural” numbers
meaningfully correspond to any physical phenomenon. They’re just abstractions
and tools for making predictions and reasoning about our experiences as
humans, which don’t necessarily reflect how the world actually is.

------
thwd
There's a joke: Mathematics is just applied Philosophy.

It bears some truth :)

~~~
dnautics
I've thought of that more of a serious thought than a joke. At my college we
had a joke that physics majors washed out and became math majors (because the
faculty were assholes); math majors washed out and became philosophy majors
(because they couldn't handle rigor); and philosophy majors washed out and
became anthropology majors (because they couldn't handle homework)

------
CalChris
Bishop Berkeley's response to Newton might qualify. Newton had something
useful but Berkeley showed he hadn't proved anything. It wasn't until Cauchy
(?) proved things rigorously that calculus was on a solid footing.

------
mikehain
Philosophy often revolves around questions of purpose in life, and any non-
mathematician might wonder why someone would devote their life to the study of
advanced math. The question of purpose becomes even more interesting when a
mathematician spends decades solving theoretical math problems that have no
immediate practical use in engineering or programming. Any mathematician who's
engaged in such problems wouldn't have any trouble understanding his own
motives; for him, the philosophy is so obvious that it goes without saying.
But an outsider might see his actions as inexplicable - as inexplicable as an
ascetic monk spending a week meditating on a frigid mountaintop.

Philosophy functions as an aid in that understanding. The mathematician might
read it and recognize truth, if it is written well, and a non-mathematician
might be able to learn the rationale for devoting one's life to math. It might
have a strong enough effect to make someone gravitate towards that type of
life.

Different works in the philosophy of math could deal with different aspects of
the mathematical life, such as the inherent beauty of an elegant proof, and it
could also go into the larger, more long-term purposes of math, whose effects
might not be recognized until long after the mathematician is dead.

I would think that the best philosophy of math would need to be written by a
mathematician. An issue with philosophy is that it's quite difficult to write
it well. There is a lot of philosophy out there that's poorly-written. It
might suffer from a lack of clarity or simply be inaccurate. To have a
brilliant philosophy of math, you would need a person who is both a skilled
mathematician and a skilled writer. That is a rare thing.

------
mymythisisthis
Most of the famous mathematicians were driven by philosophy. For example
Kepler wanted to keep perfection in the new model universe. He thought he
could do it by using Plato's solids for the relative distances between the
planets. He was wrong, but the mathematical attempts he tired help to form a
more correct answer. The motivation was philosophical.

------
platz
Also, philosophy has been dogged since Descartes with the self-imposed goal of
providing a "foundation" for all knowledge enterprises (e.g. science, math) —
by attempting to, put crudely, solve the mind-body problem and legitimatize
that our thoughts and connection to the world are valid.

------
unit91
> Question: Has it ever happened that philosophy has elucidated and clarified
> a mathematical concept, proof, or construction in a way useful to research
> mathematicians?

I would hope so! The short answer is the philosophy of Math will help you
determine whether what you're researching is _true_! Surely it would be very
bizarre to research something with complete apathy regarding its truth value.
A few examples:

The famous Peano axioms [1] are widely used to prove such things as the
commutative property of multiplication (ab=ba). But as the name "axiom"
suggests, you just have to accept them as true or the whole thing crumbles. So
why is it true that "0 is a natural number"? If this is false, much (all?) of
math research is in big trouble! Does this suggest a sort of mathematical
epistemic foundationalism? If so, what are its limits? When is mathematical
research warranted, and when can we simply regard mathematical beliefs as
properly basic?

Also, consider the realist/anti-realist debate [2, 3] which seeks to answer
the question "are numbers, sets, functions, etc. actual features of the real
world, or are they all just in our heads?" (or some refined variation
thereof). If they are real entities, how is it that these non-causal things
(like 5) lie at the very heart of the laws governing the physical, causal
universe? But if they aren't real, then what possible explanation can one give
for the perfect harmony of the physical world and these functions, that are
ultimately all in my head? Moreover, why is belief in these unreal entities so
widespread (I know of no "amathists")?

[1]
[https://en.wikipedia.org/wiki/Peano_axioms](https://en.wikipedia.org/wiki/Peano_axioms)

[2] [https://plato.stanford.edu/entries/platonism-
mathematics/](https://plato.stanford.edu/entries/platonism-mathematics/)

[3] [https://plato.stanford.edu/entries/scientific-
realism/](https://plato.stanford.edu/entries/scientific-realism/)

~~~
gizmo686
>So why is it true that "0 is a natural number"? If this is false, much (all?)
of math research is in big trouble! Does this suggest a sort of mathematical
epistemic foundationalism?

Math major here. To me, this seems like a non-nonsensical question. 0 is
_defined_ to be a natural number. You could say (as Platonism does), that
there is some "real" natural numbers out there in the universe, and ask if the
Peano axioms define a system that is isomorphic to the real "natural numbers".
Or you could ask if the Peano axioms acuratly define the artificial system
that we informally think of as the natural numbers.

You could also ask if any system (real or abstract) can satisfy the Peano
axioms. That is to say, are they internally consistent (and is there a
constructive proof of such). Assuming this is the case, then (by definition)
there will be an object that we call "0" that is a natural number. You can
also ask if there is a unique (up to isomorphism) instantiation of the Peano
axioms (there is).

~~~
unit91
> ...and ask if the Peano axioms define a system that is isomorphic to the
> real "natural numbers". Or you could ask if the Peano axioms accurately
> define the artificial system...

This is exactly what I meant. You can define anything you want. You can even
do so in a sophisticated way such that your defined system is internally
coherent. But internal coherence alone can't be the standard of measure of
truth -- we need to ask "does our internally coherent system correspond with
reality?" To the questioner on MathOverflow, _math_ will answer the coherence
question, _philosophy of math_ will answer the correspondence question.

> To me, this seems like a non-nonsensical question. 0 is _defined_ to be a
> natural number.

Is it merely definitional though? They're called "natural" numbers for a
reason! These are the set of numbers that seem most obvious to us, the kind
that (I don't think) we can teach. For example, you can tell a child "this is
1 apple", "these are 2 apples", etc., but other than the name, you really
can't teach a person what numbers are. They just know. Why is that? It's
clearly is more than merely definitional. And I think 0 falls into this
category unlike, say, complex numbers. But notice, whatever your response --
even if you disagree -- we're deep into philosophy territory here with the
simple axiom "0 is a natural number".

Regardless, thanks your your thoughtful response.

~~~
gizmo686
> For example, you can tell a child "this is 1 apple", "these are 2 apples",
> etc., but other than the name, you really can't teach a person what numbers
> are. They just know. Why is that?

I would argue that this is a question for psychology, with input from
neuroscience, biology, and likely numerous other fields of science. What you
are asking is not a question about the universe, but rather a question about
the human mind: why is it that the natural numbers seem innate to humans. In
the same way we can ask why language is innate in humans, or the skills to
walk.

Regarding 0, historically, 0 has been much more controversial as a number. To
the best of my knowledge, we have no evidence of it existing as a number prior
to around 400AD India. Also, for the record, 0 being a natural number is
denominational. So much so that many (most?) mathematicians (myself included)
do not consider 0 to be a natural number. Math has not imploded because of
this, we just waste a little bit of effort here and there to clarify what we
mean when it is an important distinction.

>And I think 0 falls into this category unlike, say, complex numbers.

Funny you should bring up complex numbers. If we look at physics, we find that
the complex numbers seem far more "natural" than the natural numbers do. In
fact, I cannot think of a single physics theory defined over the natural
numbers. In contrast, the complex numbers show up all the time. Even quantum
mechanics, which (being quantized) would seem to be an ideal candidate for a
natural number theory, ends up being defined heavily in terms of complex
numbers.

------
DanBC
It might be interesting to ask the same question on the partner site
[http://philosophy.stackexchange.com/](http://philosophy.stackexchange.com/)

------
convolvatron
if you asked practicing mathematicians whether or not the philosophy of
foundational mathematics, the hilbert programme, and wittgensteins
consternation on the notion of equality were at all relevant...you would get a
pretty dispassionate 'of course'. for whatever reasons most of these topics
are taught in the philosophy departments and not the mathematics.

------
zdean
Wouldn't mathematics simply be considered an epistemological branch of
philosophy?

~~~
ebola1717
No, not at all. One could argue that analytic philosophy and mathematical
logic have some overlap and inform each other, but even then, there are deep
differences in their goals and methods.

------
elangelcentral
Yep
[https://en.wikipedia.org/wiki/Russell's_paradox](https://en.wikipedia.org/wiki/Russell's_paradox)

------
H4CK3RM4N
I think it's worth making it clear that Plato's contributions to political
science, maths, physics, and religious thought were all framed at the time as
philosophy.

------
baq
the top answer really doesn't leave much to argue about.

~~~
bykovich
The top answer is extremely wrong.

------
joe563323
Never understood the definition of philosophy. Can philosophy be defined
mathematically ?

------
Mendenhall
hmmm I would say philosophy "clarifies" where mathematics comes from.

------
Entangled
Yes, mathematics is just an unidirectional arrow of opinionated integers.

And 42.

------
balsam
philosophy of philosophy, a godelian knot.

------
balsam
philosophy of philosophy, a godelian-gordian knot

------
bykovich
> Secondly, remember that broadly the point of philosophy is to make things
> not philosophy. In extremely simplistic historical terms, once natural
> philosophy becomes rigorous it becomes science, once philosophy of language
> became rigorous it became linguistics, and today we're seeing philosophy of
> mind turn to neuroscience.

This is absolutely not correct, and elucidates little but the prejudices of
the answerer. Philosophy of language has /not/ become linguistics, philosophy
of mind has /not/ become neuroscience, and only a subset of natural philosophy
has become natural science.

The philosophical questions discussed by Socrates have elided the grasp of
both dogmatic rigor and empiricism for twenty-four hundred years, and there
seems to be absolutely no reason to expect this to change.

The answerer has either no actual grasp of the history or content of
philosophy, or has simply decided, apparently by fiat, to discard all but the
most narrow positivism-flavored slice as nonsense.

~~~
darklajid
You seem to have a certain grasp on that subject or at least feel very strong
about it.

Unfortunately your post didn't correct any (perceived) mistake, it was mostly
a longer way to write "That's not true". Would you be so kind and expand that
reply a bit, so that I might find pointers/key words to look up myself?

~~~
4e8riufc4m9rvif
I have a beginner level interest in philosophy of mind. I can attempt to add
some information on how "...and today we're seeing philosophy of mind turn to
neuroscience." is very wrong.

Neuroscience studies various structures and mechanisms of nervous system and
brain. Even when they seemingly discuss 'consciousness', they tend to simply
mean 'awareness'. I think most are generally familiar with other aspects of
neuroscience.

In phil. of mind there are categories of consciousness or lack thereof, though
Phenomenal Consciousness is more popular area of study (experiential side of
consciousness). One of the key subject in Philosophy of mind try to show
whether mind can possibly be physical (ex readable article:
[http://organizations.utep.edu/Portals/1475/nagel_bat.pdf](http://organizations.utep.edu/Portals/1475/nagel_bat.pdf)),
can mind be described computationally? (ex paper:
[http://consc.net/papers/rock.html](http://consc.net/papers/rock.html)) Phi.
of mind also studies accesses of consciousness, study methods on how to go
about studying contents of consciousness etc.

I've talked to neuroscientist who despise philosophers talking about
conciseness and claim their ideas are silly or can be ignored. There has also
been cases where big name neuroscientist proposed silly ideas about mind that
philosophers wouldn't begin to consider.

In short they study different things.

~~~
naasking
> Neuroscience studies various structures and mechanisms of nervous system and
> brain. Even when they seemingly discuss 'consciousness', they tend to simply
> mean 'awareness'.

You should give neuroscientists more credit! Some of them are aware of the
philosophical problems surrounding subjectivity, and have tried to tackle them
[1]. That's one of my favoured scientific theories on the source of
subjectivity.

I can also sympathize with the disdain with whic some neuroscientists view
some philosophers of mind. They've debated some completely ridiculous theories
of mind in order to rationalize human importance. These consciousness debates
are following the arc of vitalism, which the science of biology eventually
simply replaced, and I expect these neuroscientists see it as a waste of time
for similar reasons.

[1]
[http://journal.frontiersin.org/article/10.3389/fpsyg.2015.00...](http://journal.frontiersin.org/article/10.3389/fpsyg.2015.00500/full)

~~~
md224
That attention schema theory is interesting but it still doesn't solve the
Hard Problem: why information about colors _looks_ like colors, why
information about surfaces _feels_ like a surface... how a biological system
could experience joy or sadness, orgasm or terror, etc. They seem to be
solving the very narrow problem of self-awareness, but not awareness in
general. And I'm pretty self-awareness isn't a prerequisite for general
awareness. (Just ask experienced drug users.)

To be fair, _no_ existing theory can explain subjective experience, so it's
not a knock on that theory specifically.

There seem to be a subset of human beings who have convinced themselves they
aren't conscious, even though it's not clear who they think they've convinced
or who did the convincing. If you're one of them, that's okay... it's just not
a conclusion I would personally endorse ("I" being whatever force is selecting
the words that you're currently reading).

P.S. If something concludes that it's aware, then is it aware it has reached
that conclusion? And if so, would that lead to an infinite regress? Perhaps
that's what consciousness is... an infinite regress of awareness. :)

~~~
naasking
> That attention schema theory is interesting but it still doesn't solve the
> Hard Problem: why information about colors looks like colors, why
> information about surfaces feels like a surface.

Because every sensation has to have some distinguishable characteristics from
other sensation for functional purposes. That's not the hardest part of the
hard problem, accounting for subjectivity was always the hard part. Certainly
there are questions that aren't fully answered, but I think this paper
demonstrates that neuroscience can and has started tackling the tough
philosophical questions.

> They seem to be solving the very narrow problem of self-awareness, but not
> awareness in general. And I'm pretty self-awareness isn't a prerequisite for
> general awareness. (Just ask experienced drug users.)

I don't know what you mean. Certainly "altered states of consciousness" alter
the operation of this attention schema theory, by altering signal strength of
some perceptions and/or impeding the function of the attention schema
apparatus somehow. I'm not sure what you think this means for this theory.

> There seem to be a subset of human beings who have convinced themselves they
> aren't conscious, even though it's not clear who they think they've
> convinced or who did the convincing.

You're grossly misrepresenting this position. I and others believe we don't
have true subjective awareness. "Consciousness" is merely a label for a
reducible phenomenon, like cars. So yes we are "conscious", but we don't mean
the same thing you mean by "conscious", which carries far more ontological
baggage that's only justified by weak thought experiments.

Finally, your cute yet all too common response begs the question by your use
of "who", ie. no one needs convincing to begin with. In fact, if we're just
automota then we're merely claiming that our perceptions yield a false
conclusion about the existence of subjectivity. That's what it means to be an
illusion.

~~~
md224
> I don't know what you mean. Certainly "altered states of consciousness"
> alter the operation of this attention schema theory, by altering signal
> strength of some perceptions and/or impeding the function of the attention
> schema apparatus somehow. I'm not sure what you think this means for this
> theory.

I was just saying that subjective awareness doesn't require awareness of the
self as observer. One can become engrossed in a movie without constantly
thinking about one's place in relation to the movie. But on second thought,
I'm probably misunderstanding some aspect of this theory, as this
counterexample seems too obvious.

> I and others believe we don't have true subjective awareness.

I'm afraid this just makes no sense to me. Aren't you subjectively aware of
the computer in front of your eyes? If you claim to not have "true subjective
awareness", I'm curious what "true subjective awareness" would amount to.

An illusion is when our subjective experience of reality does not match actual
reality, but to claim that our subjective experience is itself an illusion?
That seems like a contradiction in terms.

Anyway, I'm sorry if this is coming across as flippant. I understand where
your belief is coming from (I probably had it myself at one point) but it's
just not the way I understand the world now.

For me, the knowledge that I have true subjective awareness is a basic first
principle, along the lines of "I think, therefore I am". Maybe I'm a brain-in-
a-vat and this is all virtual reality, but I'm definitely experiencing
_something_. Are you trying to deny the fact that I have experiences and
sensations, or something else? Maybe we're just talking past each other...

Also, what "weak thought experiments" are you referring to?

~~~
naasking
> One can become engrossed in a movie without constantly thinking about one's
> place in relation to the movie.

Being engrossed involves a suspension of awareness.

> I'm afraid this just makes no sense to me. Aren't you subjectively aware of
> the computer in front of your eyes? If you claim to not have "true
> subjective awareness", I'm curious what "true subjective awareness" would
> amount to.

True subjective awareness requires ontologically committing to dualism,
because subjectivity is then irreducible. By which I mean that no account for
true first-person experience is possible using only third-person objective
facts.

> An illusion is when our subjective experience of reality does not match
> actual reality, but to claim that our subjective experience is itself an
> illusion? That seems like a contradiction in terms.

That definition of illusion begs the question, like I said, so I categorically
reject it. If you eliminate the dependency on "subjective experience of
reality" you get "perception of reality does not match actual reality", which
is exactly what I said.

> For me, the knowledge that I have true subjective awareness is a basic first
> principle, along the lines of "I think, therefore I am".

Ah, but this too begs the question! The fallacy-free version is "this is a
thought, therefore thoughts exist".

> Also, what "weak thought experiments" are you referring to?

P-zombies, Mary's room, etc.

------
hodgesrm
Has philosophy ever clarified mathematics?

Let's frame this question properly for analysis.

    
    
      All work of Aristotle is philosophy. 
      Aristotle's work includes syllogistic logic. 
      Syllogistic logic clarifies mathematical inference.
      Mathematical inference is part of mathematics.
    
      Therefore, some philosophy clarifies mathematics.
    

QED

------
Chirono
It strikes me that this is similar to the 'no true ai' paradox, where as soon
as a program can do something, such as beat a human at chess, then it stops
being AI and starts being just computation. As soon as something philosophical
becomes rigerous or well understood, we stop calling it philosophy and start
calling it mathematics.

