The Philosophy of Mathematics

[Zetetic]

No.
If so I agree that it is de facto false, given that it is useful to be able to take in to account circumstances that have a logical form requiring infinite.

Alright, we're on the same page here.

I would say it depends of what you mean by dualism. Mathematics is conceptual in nature, that doesn't mean that is physically composed out of ethereal elements.

No, it is physically composed inside of the skulls of the people who do it and recorded in a way that preserves this processing. Why do you think that the oft cited 'mathematical maturity' is? It is your ability to internalize abstract structures and develop an intuition about them. Without intensive mental recalibration, you cannot move fluently around in the more abstract areas of mathematics and mathematical logic.

If you have actually done math

I have interpreted that as an uninformed attack on my understanding of mathematics, which I take offense to seeing as it is coming from someone (an engineer) who probably has less formal education in pure mathematics than I do without further qualifying that statement. Or (B) a pointless rhetorical device designed to marginalize opinion differing from your own.

I would say that maybe if you have done -a little- math and done -a little- thinking about the process of doing it, you might be right. Your view is similar to the view I had after taking a -few- proof heavy undergraduate courses and thinking about it continually for a year or two. I have begun to strongly re-evaluate my position.

, you would know that whatever happens in the real is completely irrelevant to the propositions of math. If for instance every time you put together 1 apple with another apple in a box and you get 3 apples, 1+1 would still be 2 regardless of the physical experience that you perceive to be addition.

If you knew anything of the history and development of mathematics you would know that the vast bulk of it was derived empirically, for good reason. The propositions allow us to abstractly organize entities, including the 'abstract entities' (or the processes we use to organize once they are formalized into a set of intuitions and notational semantics) themselves. You aren't seriously trying to tell me that algebra, geometry and calculus were not developed originally to address concrete problems are you? Are you maintaining that the mathematical formalisms based on these fields was not developed by mathematicians who empirically experienced older mathematics and developed way to organize the ideas further? Or do you just not consider the act of reading and internalizing the material in a mathematics book to be empirical despite it's similarity with empirical observation?

Also, do you really think that the processes that combine to form the meta-process of doing mathematics are not real processes nor are they related to them? "you would know that whatever happens in the real is completely irrelevant to the propositions of math" .......uh huh.....

One way that I can interpret your statements is that they are anti-scientific and ignore neuroscience and cognitive science entirely. Mathematics can be represented as a process in the brain, to say otherwise is to contradict the very idea that we could map our thoughts in any meaningful way physically. This is not only dualist (who generally accept that there is likley a perfect correlation between patterns of thought and patterns of the electrochemical stimulation in the brain).

Another way indicates to me that you mean that there is nothing external to the brain that influences mathematical computation (this is clearly wrong, as the very act of learning mathematics is interaction with the outside world in a way that affects your mathematical thinking and the way that you process mathematical ideas).

I will give you the benefit of the doubt here and assume that you failed to make your ideas fully clear. I assume that you mean that the logical validity of a proposition is not contingent upon whether an apple falls from a tree in Florida, because we define the valuations of our variable sets in a way that, aside from applications, is unrelated to external events. This is not even tangentially related to New Empiricism though, so I don't see the relevance. If you ask me, I would say that you didn't read the entry on New Empiricism with much care and simply assumed it was coming from the same place as Empiricism proper, a mistake.

... But the propositions found in math is completely independent of any socio-economic circumstance, the fact that there is one and only one decomposition in prime factors of any positive integer would still be true regardless for instance if you believe in Jesus or not.

No shit, which is why I didn't say that it wasn't and in fact I explicitly indicated this. "Therefore there are absolutely sociological aspects to the progression of mathematics (though the affect on actual mathematical processing is unclear at best)." You might not have understood what I meant here; I am saying that there is little to absolutely no 'soci-economic' (as you call it, though that probably isn't how a sociologist would phrase the issue and is not really specific enough to address their claims, which I nevertheless maintain are wrong) influence on how the human brain computationally approaches mathematical problems.

Sorry to tell you this, but it is not only useful as it is also the only way we do math, if it is not abstract it is not math period (it is one of the basic properties that makes math math).[\quote]

I should have qualified this statement: I don't consider visualizing mathematical systems to be synonymous with Platonism. Every aspect of how we do mathematics is located in our brains, which are essentially quantum universal Turing machines (maybe extended in some ways we haven't fully conceptualized). There is no utility in declaring mathematical systems to be 'real abstracts', what is relevant to us is how a mind can physically compute the mathematical statements and how this computation is experienced by us.

Also wtf is this: "if it is not abstract it is not math period (it is one of the basic properties that makes math math)." That is so vague as to be practically meaningless. You don't present a coherent meaning to 'abstract' as it relates to mathematical processing, you don't qualify what it means for an object to 'have the property of abstractness', I'm not sure why I should not interpret this as the result of a muddled view of mathematics and it's relation to the physical world.

As far as what they are physically : mathematics and logic are processes that take place in the brain. They probably require a universal quantum Turing machine to do (though this may be only one component), and the way that humans process them is possibly going to be distinct from the way that another sentient being would, because we tie the formalisms to our intuition and we cannot expect another species to have the same intuition as humans, since their brains would likely be structured differently and respond differently to perceptual cues. That is not to say that they would not recognize our mathematics as such, they would just intuit it slightly differently, though their computations should lead to the same results.

Again, no. It is not just generally correct, nothing in mathematics is established unless it is formal, you can dwell in "informal" ideas in order to plan a line of reasoning to follow but you will never establish anything unless you "walk the formal path".
So in essence your ideas on mathematics are completely upside down.

You have basically repeated what I have said in your response.

I think your comprehension of my post is completely upside down (though I am willing to grant possible language difficulties and possibly that you didn't even bother to read it carefully). I've taken several graduate courses in mathematical logic, computer science, Algebra and Analysis (real and complex) and have done some (though not much, I'll admit) original research in mathematics so forgive me if I don't take your seemingly glib approach to what I am saying very seriously. I think that if you read what I actually said with a little care you would realize that you are skimming and giving a very non-charitable interpretation of what I am indicating.

"Formalism gives an incomplete picture" ........"nothing in mathematics is established unless it is formal, you can dwell in "informal" ideas in order to plan a line of reasoning to follow but you will never establish anything unless you "walk the formal path"."

::sigh::

If you paid attention to the context, what I meant was that Formalism does not give a complete picture of what is being done when people DO mathematics. You admitted so much when you said Platonism is useful to mathematicians.

This is why someone has to develop mathematical maturity and intuition (which is what I think many mathematicians conflate with Platonism) before he can parse the formalisms involved in an abstract text.

In effect, all that I am stating is that the brain does not process mathematics in the same way a Turing machine based automated theorem prover does. It ties things back to other areas of human intuition in order for us to gain a fluency with the formal systems we encounter.

 

[Master_Ghost_Knight]

No.
If so I agree that it is de facto false, given that it is useful to be able to take in to account circumstances that have a logical form requiring infinite.

Alright, we're on the same page here.

ARGH! No. I have only intended to say No. The rest of the setence was meant to be part of the a next quote that I taught I have removed. Those are not my words (they are yours piece by piece), sorry about that because I disaree with the statment.

No, it is physically composed inside of the skulls of the people who do it and recorded in a way that preserves this processing. Why do you think that the oft cited 'mathematical maturity' is? It is your ability to internalize abstract structures and develop an intuition about them. Without intensive mental recalibration, you cannot move fluently around in the more abstract areas of mathematics and mathematical logic.

Just becaue you need a mature brain to understand mathematics it doesn't mean that the propositions of mathematics change if your mental circumstance change, in fact that doesn't happen.

If you have actually done math

I have interpreted that as an uninformed attack on my understanding of mathematics, which I take offense to seeing as it is coming from someone (an engineer) who probably has less formal education in pure mathematics than I do without further qualifying that statement. Or (B) a pointless rhetorical device designed to marginalize opinion differing from your own.

Just because I am an engineer, it doesn't mean that I don't have a good education in mathematics, quite the oposite in fact, engineers are the 3rd most proficient in the field of mathematics (behind pure mathematicians and theoretical physicists). And as in a matter of fact more engineers formed in my University are called in to teach math due to their proficiency then actual mathematicians from other Universities (just so you have an idea).
But the fact is to be able to understand the position I am talking from you would need to go trough the same level of education that I have (which from the way you talk I know you haven't been anywhere close), how am I going to convince you that you must necessarily reach the exact same conclusion for every single mathematical establishment ever made with logic alone (whithout showing you all that being done in front of your eyes) regardles of anything else you whish to consider? I don't have time to teach you good math, so in away you got to take my word for it in the basis that neither of us has the intention to lie with the only difference that I have actualy been trough the experience.

I would say that maybe if you have done -a little- math and done -a little- thinking about the process of doing it, you might be right. Your view is similar to the view I had after taking a -few- proof heavy undergraduate courses and thinking about it continually for a year or two. I have begun to strongly re-evaluate my position.

Really? Can you give me an example of a mathematical establishment that would change its validity either due to empiric experience, socio-economic status or any other subjective circumstance? Or do you think that such is even possible?

, you would know that whatever happens in the real is completely irrelevant to the propositions of math. If for instance every time you put together 1 apple with another apple in a box and you get 3 apples, 1+1 would still be 2 regardless of the physical experience that you perceive to be addition.

If you knew anything of the history and development of mathematics you would know that the vast bulk of it was derived empirically, for good reason.

This statment is false, it goes contrary to everything we actualy do know about mathematics and the history of it.

The propositions allow us to abstractly organize entities, including the 'abstract entities' (or the processes we use to organize once they are formalized into a set of intuitions and notational semantics) themselves. You aren't seriously trying to tell me that algebra, geometry and calculus were not developed originally to address concrete problems are you?

Yes, they where not all developed to adress concrete problems, the vast majority of it is persued for knowledge sake independently if you can find a physical use for it later. It is indeed true that some of the mathematical avenues where preferentially developed with the purpouse of being used in physical problems, but it is absurdly false to claim that if those physical problems didn't exist that then you would be unable to establish those same statments, much more absurd that those same statments could have been false if the physics said otherwise. You got it reverse my friend, math tells physics what is true and never the other way arround.

Are you maintaining that the mathematical formalisms based on these fields was not developed by mathematicians who empirically experienced older mathematics and developed way to organize the ideas further?

Just because mathematics had an early stage like Astronomy had Astrology and Chemestry had Alchemy, it doesn't mean that it can't be completly divorced from it. It may indeed be true that in acient history people learned how to count with sheep, but todays mathematics is completly diferent, reformalized and divorced to the very core from what was precieved as mathematics in acient times. Even the simple task of counting (that appears undistinguishable on the surface) is completly different from that of what you used to count sheep. Never in history a sheep hearder had to worry about "having to justify the existance of a set with a set of axiomatic operands with propreties like the unicity of the neutral elements, propreties like the distributivity, associativity and comutativity and the distinction bewteen the 2 neutral elements from the 2 operands" in order to establish that he is 2 sheep short if has 48 in the curral when he previously had 50.
And if you think otherwise you are heavily uneducated to say the least.

Also, do you really think that the processes that combine to form the meta-process of doing mathematics are not real processes nor are they related to them? "you would know that whatever happens in the real is completely irrelevant to the propositions of math" .......uh huh.....

One way that I can interpret your statements is that they are anti-scientific and ignore neuroscience and cognitive science entirely. Mathematics can be represented as a process in the brain, to say otherwise is to contradict the very idea that we could map our thoughts in any meaningful way physically. This is not only dualist (who generally accept that there is likley a perfect correlation between patterns of thought and patterns of the electrochemical stimulation in the brain).

What you lack is a sever understanding in philosophy. So here is lesson one of logic. The establishment of the truth value of a statment is only guaranteed if the statment is a necessary result from truth premessis. If the world is in such a way to impair your cognitive capabilities to the extent that your conclusions of truth value can not be based on truth pemisses, then you can not guarantee the truth value of any statment including any statment that by nature would requier the violation of this statment. As such rendering you incapable to truly kow anything. Since this version of the world can be true or not be true, it is therefore only usefull to assume that such version of the world is not true given that if it were indeed true then any statment from here on end is inconsequential by definition.
Sumarizing, you can only sucessfully use logic if you can use logic (painfully tautoligical), if you can not, then it doesn't really matter because there is nothing that can guarantee that anything you do outside logic is true, making it useless. Do you understand now why your position is fundamentaly absurd?
Let me further this one notch. The fact that you were able to establish that mathematics is performed in the brain is dependent on logical basis and most of them requiered math, if you can not guarantee the validity of math you also can not therefore guarantee the valid of the statment that math can be mapped on the brain, even before you are able to guarantee that math is faulty because the brain is faulty. Logic is a bitch ain't it?

Also wtf is this: "if it is not abstract it is not math period (it is one of the basic properties that makes math math)." That is so vague as to be practically meaningless. You don't present a coherent meaning to 'abstract' as it relates to mathematical processing, you don't qualify what it means for an object to 'have the property of abstractness', I'm not sure why I should not interpret this as the result of a muddled view of mathematics and it's relation to the physical world.

If you can find me a 4 in the real world then I give it to you that it is real, otherwise it is abstract in the sense that there is only the concept of a 4 for wich there is no fiting precieved physical paralel. What the hell do you think abstract means? Can you give me an example of something in math wich isn't abstract? Are you seriously to stupid to understand the concept of abstract?

I think your comprehension of my post is completely upside down (though I am willing to grant possible language difficulties and possibly that you didn't even bother to read it carefully). I've taken several graduate courses in mathematical logic, computer science, Algebra and Analysis (real and complex) and have done some (though not much, I'll admit) original research in mathematics so forgive me if I don't take your seemingly glib approach to what I am saying very seriously. I think that if you read what I actually said with a little care you would realize that you are skimming and giving a very non-charitable interpretation of what I am indicating.

I doubt that very much sense you yourself have stated that mathematics is empirical, math has no such proprety. Do you prove the pitagorian theorem by drawing triangles on a piece of paper? If you have done actual math, How dare you say that?

If you paid attention to the context, what I meant was that Formalism does not give a complete picture of what is being done when people DO mathematics. You admitted so much when you said Platonism is useful to mathematicians.

Being formal and platonic are not 2 mutualy exclusive propreties you know.

 

[Zetetic]

I don't think that you have understood anything I have said. If you want a clear idea of a position that is similar to mine, you should read this http://www.peirce.org/writings/p119.html and this http://www.peirce.org/writings/p107.html.

Peirce was an eminent scientist and logician.

Now, I refuse to separate mathematics from the class of physical phenomena that allow it to be done. I do not see a good reason to do this. This is effectively what I mean when I say that mathematics is empirical, I mean that we learn it by reading books and gaining intuition. In a sense, everything is empirical by virtue of how we interact with it and update our expectations based upon our interactions.

I can only describe the courses I have taken in mathematics and say that I have studied philosophy intensely for several years. I have admittedly only take two graduate courses in mathematical logic, covering up to Godel's first and second incompleteness theorem. I fully understand the theorems, and how they relate to incomputability proofs. I am currently taking a graduate seminar in Computational Complexity theory from a professor who was a student of Kleene who to some extent shares my convictions. I am also taking a graduate sequence in Measure theory. I am fully familiar with Solomonoff induction and am learning algorithmic information theory presently. I have studied Algebraic Geometry and Concrete Algebraic geometry and have done research on applications to Groebner bases to post quantum cryptology.


My position is very similar to that of Peirce's pragmatism. It is a difficult position to understand, especially if you have been conditioned to Logical Positivism (which is the position I used to hold, because of it's elegance and seemingly correct ideas) but I think that the two articles I have given you will assist you should you choose to endeavor to try to actually understand my position. Previously I was captivated by the Philosophies of A.J. Ayer and Carnap and early Quine, but I found them to be lacking in an essential way.Wittgenstein took my interest but I rejected him because he was too vague and Peirce's position added much clarity to the intuitions I have gleaned from my reading of Wittgenstein. I also had a brief brush with Krpike's ideas about how to interpret Wittgenstein, and he is interesting on the subject.

All I can tell you is that you should read the two articles, reread what I wrote and get back to me. Pragmatism is increasingly prevalent in modern philosophy, and it takes quite a bit of thought to avoid being instantly put off by what they have to say, but I can assure you that if you put in the effort to really truly understand the position, you will be better for it.

 

[Master_Ghost_Knight]

I don't think that you have understood anything I have said. If you want a clear idea of a position that is similar to mine, you should read this http://www.peirce.org/writings/p119.html and this http://www.peirce.org/writings/p107.html.

Peirce was an eminent scientist and logician.

Holy crap, you got me there. That was the most painfull reading I have ever had in a while, the guy meaders arround in pointless dialog never ever making a conclusion or telling you anything in concrete really, and spews out his life story like there was no tomorow, more akeen to a romance rather then a philosophycal dialog. In various ocasions I felt like beating the crap out of the author (failed there this being the internet) and tell him to get to the fucking point, which sadly there wasn't any. Very ironic really since this was supoused to explain the importance of clear ideas, well so much for that I guess. And the parts that were more or less bareable, he goes arround and says "After this tedious explanation, which I hope, may not have exhausted the reader's patience".

Ok so here is what I got. The guy commits many logical falacies, including guessing what the hell proiminent personalities dead in 18 century might have taught and basing a theory on that to prove his case. Right! Let's go team retard!
Sorry to tell you this, it is pseudo-intelectual rubish.

For the rest of your comments all I have to say is this. What you personaly feel about the issue is completly irrelevant to it, refusing to accept the logical answer in favour of that which pleases you is a logical falacy, and I am dismissing it as such.
You don't know what empiric means, and you hold the absurd idea that you are unable to think (aparently you got that from Pierce), so the obvious question comes to mind, where did the knowledge we talk of came from? You might want to check that out.

 

[Zetetic]

A few questions about your little rant:

[
Holy crap, you got me there. That was the most painfull reading I have ever had in a while, the guy meaders arround in pointless dialog never ever making a conclusion or telling you anything in concrete really, and spews out his life story like there was no tomorow, more akeen to a romance rather then a philosophycal dialog. In various ocasions I felt like beating the crap out of the author (failed there this being the internet) and tell him to get to the fucking point, which sadly there wasn't any. Very ironic really since this was supoused to explain the importance of clear ideas, well so much for that I guess. And the parts that were more or less bareable, he goes arround and says "After this tedious explanation, which I hope, may not have exhausted the reader's patience".

If you want a more formal account of Peirce's philosophy of mathematics that is more direct and contrasts it with other ideas in the philosophy of mathematics you can look at chapter 2 and 3 of this : http://www.helsinki.fi/~pietarin/publications/Pragmaticism-Antifoundationalism-Pietarinen.pdf

Ok so here is what I got. The guy commits many logical falacies, including guessing what the hell proiminent personalities dead in 18 century might have taught and basing a theory on that to prove his case. Right! Let's go team retard!
Sorry to tell you this, it is pseudo-intelectual rubish.

So you contend that referencing philosophers in a work of philosophy is 'retarded'? Just trying to see if I understand you correctly.

I'm not sure which logical fallacies you are referring to, you should be able to be more specific.

Here is an account of Peirce's mathematical and logical accomplishments if you are not convinced he knew how to do either logic or mathematics:http://en.wikipedia.org/wiki/Charles_Sanders_Peirce#Mathematics

For the rest of your comments all I have to say is this. What you personaly feel about the issue is completly irrelevant to it, refusing to accept the logical answer in favour of that which pleases you is a logical falacy, and I am dismissing it as such.
You don't know what empiric means, and you hold the absurd idea that you are unable to think (aparently you got that from Pierce), so the obvious question comes to mind, where did the knowledge we talk of came from? You might want to check that out.

I am unsure of where you got the idea that I hold that I am unable to think. Could you be more specific? What did I say, what sentence did I type out, that made you think that I hold this absurd notion? This only strengthens my conviction that you are for some reason unable to specify what I am being unclear about and substitute in your own assumptions about what I mean.


I think I have found a paper that describes my views better than Peirce does, though I am not totally convinced it is as complete as Peirce's ideas (which I am testing out by arguing here). Here it is :http://philsci-archive.pitt.edu/1164/1/formfiz_preprint.pdf It is a physicalist account of mathematics.

 

[Master_Ghost_Knight]

If you want a more formal account of Peirce's philosophy of mathematics that is more direct and contrasts it with other ideas in the philosophy of mathematics you can look at chapter 2 and 3 of this : http://www.helsinki.fi/~pietarin/publications/Pragmaticism-Antifoundationalism-Pietarinen.pdf

Much berable read, it actually conveys properly information. I don't have time to read it all tough (I will do it later), but from I did read, the author confuses sometimes math with natural siences despite the fact that he was warned about it. There are some points I disagree (in what I have read so far) even tough he presented a view of mathematics which are contradictory to his opening statments (openly admiting that it is neither empiric or social), but I will summarize my opinions on it later.

So you contend that referencing philosophers in a work of philosophy is 'retarded'? Just trying to see if I understand you correctly.

What is retarded is to name some philosophers (mathematicians), make explicit unfounded assumptions on what they might have taught and establish a conclusion from there, in a argument from authority kind of way.

For the rest of your comments all I have to say is this. What you personaly feel about the issue is completly irrelevant to it, refusing to accept the logical answer in favour of that which pleases you is a logical falacy, and I am dismissing it as such.
You don't know what empiric means, and you hold the absurd idea that you are unable to think (aparently you got that from Pierce), so the obvious question comes to mind, where did the knowledge we talk of came from? You might want to check that out.

I am unsure of where you got the idea that I hold that I am unable to think. Could you be more specific?

The fact that you have said that knowledge of mathematics is essentially empiric and not platonic, (and by the painfull words of pierce) that math is learned by external experiences, meaning that you almost exclusively know if you see it happen and you are essentially unable to formulate ideas which have not been experienced. Th real picture is quite the oposite.

I think I have found a paper that describes my views better than Peirce does, though I am not totally convinced it is as complete as Peirce's ideas (which I am testing out by arguing here). Here it is :http://philsci-archive.pitt.edu/1164/1/formfiz_preprint.pdf It is a physicalist account of mathematics.

I wll read it later and comment on it.

 

[Master_Ghost_Knight]

This is what I got: The first paper is a good read, I take issues with the fact that he is passing judgement on what he thinks someone else meant but that is beside the point. The paper seams a bit inconsistence sense he makes the opening statment remarking one hing (which I generaly heavily disagreed) and during the body of the paper he instead expresses the oposite (which I generaly agreed, but inconsistence is noticed).

The second paper is farily redable but it obviously come from someone completly ignorant of mathematics and very often confounded it with physics, ending out supporting that math is just like any other natural science that is under falible subjective experiences and interpretation (sofice to say that it is nonsense).
Sticking to the later rather than the former only further my convictions that I have correctly interpreted your point.

 

[Zetetic]

Much berable read, it actually conveys properly information. I don't have time to read it all tough (I will do it later), but from I did read, the author confuses sometimes math with natural siences despite the fact that he was warned about it. There are some points I disagree (in what I have read so far) even tough he presented a view of mathematics which are contradictory to his opening statments (openly admiting that it is neither empiric or social), but I will summarize my opinions on it later.

Where in the introductory statements are you seeing him claim that mathematics is empirical or social? As far as I can tell, in chapter 2 and 3 he is arguing about what Peirce's mathematical philosophy was as opposed to other viewpoints, not about which was correct.

What is retarded is to name some philosophers (mathematicians), make explicit unfounded assumptions on what they might have taught and establish a conclusion from there, in a argument from authority kind of way.

That's not what he was doing, he is assuming that the reader is fully familiar with the ideas of those philosophers and is arguing against the conventionally correct interpretation of them. The underlying assumption is that, since it was publish in a journal of philosophy, that the audience is fully aware of the canonical interpretations of the philosophers he mentions, and that they are capable of evaluating his arguments in relation to the claims made by those previous philosophers.

Seriously, have you never read any serious piece of work in philosophy written in the last three hundred years? Even Rousseau did this! The set compliment to the set of your knowledge is co-finite to it with respect to the sum of human knowledge! (ok, that was hyperbolic, the set of human knowledge is finite).

The fact that you have said that knowledge of mathematics is essentially empiric and not platonic, (and by the painfull words of pierce) that math is learned by external experiences, meaning that you almost exclusively know if you see it happen and you are essentially unable to formulate ideas which have not been experienced. Th real picture is quite the oposite.

What a contorted and broken piece of reasoning on your part. You are able to form ideas THAT THE TOTALLY PHYSICAL BRIAN IN YOUR HEAD CAN FORMULATE. THE FORMATION OF IDEAS IS A PHYSICAL PROCESS. THIS IS ALL THAT I AM SAYING. Let me say that your skull shares a property with the set of rational numbers and I'll give you a hint: (this is when LateX could come in handy) For all A < B in the set of rational numbers there exists a number C such that A < C < B .

This is what I got: The first paper is a good read, I take issues with the fact that he is passing judgement on what he thinks someone else meant but that is beside the point. The paper seams a bit inconsistence sense he makes the opening statment remarking one hing (which I generaly heavily disagreed) and during the body of the paper he instead expresses the oposite (which I generaly agreed, but inconsistence is noticed).

What do you mean 'passing judgment', what rubbish. Philosophy is done relative to other philosophy, it is rather like a rigorous discussion, not unlike what we are doing here. He is expatiating on what he considers to be Peirce's ideas on the subject and supporting it with evidence in the form of quotes from his published works. This is what you consider to be retarded?

Where do you get the idea that the author is making the claim that mathematics is social and empirical?

The second paper is farily redable but it obviously come from someone completly ignorant of mathematics and very often confounded it with physics, ending out supporting that math is just like any other natural science that is under falible subjective experiences and interpretation (sofice to say that it is nonsense).
Sticking to the later rather than the former only further my convictions that I have correctly interpreted your point.

[/quote]

I had not fully read the paper, I admit. I disagree that mathematics is fallible, but it must take place entirely by physical processes. I do not think that this means that therefore it is contingent on future observations or social whim. I agree that mathematics is absolute and have done so from the beginning.

I hold that mathematics is essentially structural, that is, it only contains information about how things can be in logical or spatial etc. relation to one another, and two isomorphic structures with identical representations and interpretations (in the model theoretic sense, which is very precise) are mathematically the same (though they may use different symbols). I also hold that the act of doing mathematics is strictly a physical process that can only be done with physical tools (some form of which are present in the brain).

Let me present my view of Idealized science and empiricism here: We formalize our expectation about a physical system as a hypothesis. We devise a body of facts that should be the case if our hypothesis is correct. We now have a Bayesian evidential model of our system. As we test each expected fact that should be entailed by our hypothesis and find them to be true, we have a higher and higher probability of correctness. If we encounter a prediction that is false but necessitated by our hypothesis we are forced to reject the hypothesis.

Now, in practice it is essentially impossible (outside maybe quantum mechanical systems) to devise every contingent fact that should be true if our hypothesis is correct, but we can send ideas through the battery until we have quite a bit of evidence for them with no evidence to the contrary.

Now I should state my view of the relationship between mathematics and science: Mathematical structures aid our intuition about physical structures and systems because they give us various logics of interaction; that is, because mathematics is the study of possible logical and spatial relationships, as long as we can represent physical relationships as logical or spatial ones, we can use our mathematical intuitions and knowledge in order to organize our physical knowledge. The mathematics alone, however, is incapable of discerning whether it fits the physical systems we are applying it to, which is where we need Bayesian inference and human intuition.

So I hold that the brain is an engine that can compute the possible logical and spatial relationships possible actual objects can hold with respect to one another, and that if there are some physical contingencies related to mathematics, they are only related to whether the engine for producing the mathematics is running properly (which is a physical fact that could be true or false).

I hold that there is a totality to what possible physical objects can display in terms of their relationships to one another, and that therefore there is a totality to mathematics. In a sense it is empirical, in that we must discover so called 'a priori' truths by a physical process of investigation and we must imagine possible objects and hypothesize about what would be the case if they followed certain rules, if certain information were true about them. We might postulate without proof in mathematics, but there is a clear way of knowing whether our hypothesis is correct with certainty rather than probabilistically, as we have to make do with when asking the question of science, which is : "Does this mathematical structure fully describe this physical one?".

So any engine that can compute logical and spatial representations of possible actual objects and systems will eventually come to the same conclusions with the same starting conditions.

I hope now that I have removed speaking in 'isms' from the representation of my point of view that you understand it.

 

[Master_Ghost_Knight]

I don't have time to re-read it and get you the actual quotes, but it did occured to me sometimes that the author said "it surprises me that X wasn't Y (...paragraphs later) X classified it to have all the propreties of Y". That is secondary, I will not insist in that.

Regarding the details of people telling you the way other people think, you can not know what other people think, at best of what you can and should have done is to remark what the aformentioned peson expressed. If he could relace the "he thaught" with "he said" while forming a factual statment (in the sense that he has actualy said it and was not simply interpreted from his work like Peirce openly admited to assume that he taught), then I wouldn't be raising this objections (I think I need not explain why it is an imprtant objection).

What a contorted and broken piece of reasoning on your part. You are able to form ideas THAT THE TOTALLY PHYSICAL BRIAN IN YOUR HEAD CAN FORMULATE. THE FORMATION OF IDEAS IS A PHYSICAL PROCESS. THIS IS ALL THAT I AM SAYING. Let me say that your skull shares a property with the set of rational numbers and I'll give you a hint: (this is when LateX could come in handy) For all A < B in the set of rational numbers there exists a number C such that A < C < B .

Well this isn't empiricism. My objection regarding this is not one of fact but of relevance, to express that mathematics has the propreties A, B and C in a constructivist matter we must not forget when applying new data what should take precedence and what should influence the propreties atributed to it. It is indeed true that mathematics is a process that takes place in the brain which is also responsible for every taught process, but we are not born knowing that, that is not something that we know for certain to be necessary like for instance the law of excluded midle (i.e. something can not both be and not be at the same time and in the same context). There is an assumption being made at the very core in order to use logic and acquire knowledge, not one of fact but one of relevance, and this assumption is that you can establish reliable conclusions whitin certain rules (mathematics doesn't requier any other assumptions other than those made to perform logic in order to ensure reliability). Now we haven't made the same assumption regarding the senses, not only because the senses have decieved us in the past, but also because reasoning is the only means we have to acess acurracy, notice that we take reasoning over senses is not a statment of fact but of relevance.

At this point you have 2 worlds (sort of a dualism), one of reason and one of senses, and it is from this prespective that we explore the physical world. Falibility of the physical knowledge comes primarly from the fact that the information we get from the physical world comes to us trough senses (which we have already established to be a falible chanel) and we try to acess atributes in the physical world. But when we learn for instance logic or math from a book, even tough the information comes trough the senses what we do is to reconstruct that information in mental entities with a set of propreties capable of being processed by reason (that we call abstract) as if those ideas were there all along. Now if those ideas with those set of propreties after going trough the reasoning process happens to match the reconstructed information of what you should expect to get (when those ideas go trough a specific reason process), the you say that this ideas that come from the senses are "reasonable" (if they don't match then they are not reasonable). It didn't matter that the information has come from the falible senses, not also because an idea is either valid or not no matter were it originates but also because you have reconstructed the ideas in your mind to pass judgement on the by-product of those ideas in that reasoning process in the plane of the world of reasoning (which we have alredy assumed to be reliable). So despite the fact that information originated from the senses (empiric fashion) the ideas themselves where reconstructed on the mind and has only been processed with reason, the outcome of that reasoning process given that idea did not depend on aspects of the physical world and therefore the conclusion was not empiric (even if the idea arrived trough empirical means).

Of course with the development and understanding of science (which is an assesment of the world beyound our senses) we have come to realise that the world of reason and the world of the senses are not 2 seperate things at all, but let's not forget that this information has come from the falible senses and we are passing judgement on the worldof the senses (which we can be wrong). The implications of this is an assesment of the physical world (therefore not reliable) which says that because the physical world is the same has the reason world (as precieved trough the senses) therefore the reason world is not necessarilly reliable. This is not a completly unreasonable acessment to make, however the necessrilly is a statment of probability meaning that in truth in can be or not be, but we have already made the assumption at the biggining that the world of reason is reliable way before we made this acessment. This acessment is dependent on the reason being reliable, if it is the oposite then you can neither ensure the reliability of this statment (much less because it is an acessment that we know is not reliable), but note, the assumption of reliability of reson was not one of fact but of relevance. Given this what takes precedence? And given this would it after all that change any proprety we already atribute to process of reason?

 

[Zetetic]

Of course with the development and understanding of science (which is an assesment of the world beyound our senses) we have come to realise that the world of reason and the world of the senses are not 2 seperate things at all, but let's not forget that this information has come from the falible senses and we are passing judgement on the worldof the senses (which we can be wrong). The implications of this is an assesment of the physical world (therefore not reliable) which says that because the physical world is the same has the reason world (as precieved trough the senses) therefore the reason world is not necessarilly reliable. This is not a completly unreasonable acessment to make, however the necessrilly is a statment of probability meaning that in truth in can be or not be, but we have already made the assumption at the biggining that the world of reason is reliable way before we made this acessment. This acessment is dependent on the reason being reliable, if it is the oposite then you can neither ensure the reliability of this statment (much less because it is an acessment that we know is not reliable), but note, the assumption of reliability of reson was not one of fact but of relevance. Given this what takes precedence? And given this would it after all that change any proprety we already atribute to process of reason?

That physical discovery is fallible does not mean that it is false, so we cannot so easily brush it aside when our process of physical discovery claims that mathematics might be fallible in at least as far as that it is physical. It may be that mathematics is a real general, like the physics of gravitation seems to be (relative to the inertial frame the field is in). So then we might have some uncertainty about it's truth in an absolute sense, but it has not failed us in any way yet, and is so powerful that we cannot even conceive of how it might fail us in the future if such a failure is possible, though we can worry about what such a failure might mean after we fully understand it physically.

No appeal to tradition can aid us here. We cannot give credence to a false and ancient dogma due solely to its primacy and popularity. I hold that mathematics is an entirely physical process and that only by understanding it physically can we dissolve the issue further. I think it likely that if we study it in neuroscience and neurophysiology and biochemistry, then we will at the very least have a much clearer picture of what we are doing in terms of concrete physical operations when we do mathematics. I think that this will likely be sufficient to clear up the current level of the conundrum, though it may give us deeper problems.

At the very least, such a line of scientific inquiry could give us hints about how we might improve our ability to do mathematics and think logically, so it would be good in any event.

 

[Master_Ghost_Knight]

That physical discovery is fallible does not mean that it is false, so we cannot so easily brush it aside when our process of physical discovery claims that mathematics might be fallible in at least as far as that it is physical.

I don't think you have read anything I said.

It may be that mathematics is a real general, like the physics of gravitation seems to be (relative to the inertial frame the field is in).
So then we might have some uncertainty about it's truth in an absolute sense, but it has not failed us in any way yet, and is so powerful that we cannot even conceive of how it might fail us in the future if such a failure is possible, though we can worry about what such a failure might mean after we fully understand it physically.

You haven't read anything, and once again you fall into the pitfall which you yourself claimed that it did not applyed to you when I have layed out my criticism of it.
Do you understand any of the implications from the propreties I have mentioned and how it applys to what you are saying?
I will not bother wasting my time with you if you are not one least bit worryed about adressing any of my points and my only role here is to simply tell you to go read it again. There are obvious textbook examples on how you could put objections to my reasoning (and I have already prepared to explain why they are not, because it was so obvious to me that some one would inevitably raise them that I might as well save time) and I taught we could actualy have an interesting discution (because it is an interesting topic), but no, you rather keep to fundationaly flawed ideas already adressed and commiting the falacy of confusing the process with the product (How is this not obvious to you?).

At the very least, such a line of scientific inquiry could give us hints about how we might improve our ability to do mathematics and think logically, so it would be good in any event.

The ability to do math is different fom math. If you want an idea how "math can fail" (you could have raised this, it is a good point, but it never hinted you) everyone has had the fortune of misscalculating something, of course the problem here is that math only guarantees acurrate results if you follow it's rules, if you missed a couple of those rules leading you to a mistake you are not really doing math. And you could explore this avenue elegantly, but no, you rather build your wok from something you don't know instead of something you do know. So what can I do, maybe it is my fault that I'm not making my case clear.

 

[Zetetic]

This isn't getting anywhere, so I'll try a new approach.

I was considering the process of modeling a mathematical problem or object or class of objects as a logical clause or recursive set of such clauses, for instance, as we have in the case of Peano arithmetic. There is a research program called 'Reverse Mathematics', and it is interesting in and of itself, but what makes it really interesting is the fact that we can run into theorems that are to us intuitively correct, but do not follow from the axioms we have devised. For instance, we might like that the Continuum Hypoethsis be correct, but it turns out that it is independent of the ZFC axioms, and a great deal of resources over a century had been devoted to trying to prove it to be correct or false. It was not until Dana Scott developed his forcing technique that the problem could be settled, and it was not intuitively obvious to any but a few mathematicians that an independence proof was needed until the later half of the twentieth century.

The issue arises from the act of fitting our mathematical intuitions to logical propositions. This is not so clear cut a process, and when we have failed to do it properly the results are not always instantly clear (as seen with the CH). What this highlights to me is that, especially when the mathematical framework becomes very complex, it is difficult to tell when we have captured it logically. Reverse mathematics is a research program that tries to classify different mathematical theorems by the axioms required to prove them. http://en.wikipedia.org/wiki/Reverse_mathematics

My question then, is how we can reliably translate our mathematical intuitions to our logical formalisms? This problem seems to me to indicate that mathematical intuition comes from a different set of thought patterns than the formalisms we use, though it might just be because we have not sufficiently gained faculty in all of the intricacies of the logical frameworks we are using, and so they appear distinct, or maybe that our intuitions are confused, and we fail to really propagate our meanings and are trying to hold subtly contradictory ideas together.

What is it that is going on? We can certainly break it down: There is a mathematical structure that we are considering, and we would loike to use logical formalisms to express it. We consider features of this mental object and encode them in the notational schematic of formal logic, we might build a model of the system we wish to consider. What does it mean when we argue that some of our intuition is not captured by the formal system?

In a trivial case, we have only a small amount of relevant information, one such trivial case is the bridge of Konigsberg problem, where we only needed information about the graph theoretic aspect (which Hilbert developed quite impressively, as Hilbert is want to do) to answer the question about the bridges, which is whether we can cross all of them without passing any one of them twice.

In a more complex case we have Geometry. During the Enlightenment everyone assumed (and it is a very convincing seemingly a priori gut instinct that drives this) that Euclidean geometry was the only logically correct way of looking at spatial relations, and of course they were wrong. There are multitudinous ways of considering how objects can be organized spatially, of which Euclid's axiomatization is one. So by constructing a different set of axioms by which we can organize spatial information we can have our very intuitions shifted.

The trouble I have then, is that we seem to have this factor of intuition that seems to be highly malleable and what we seem to be doing when we are creating new mathematics rather than building on old formalisms, is somewhat distinct from what we do when we are logically deriving new axioms. In fact, it seems more like what we do when we fit a logical model to a real situation and try to derive truths about it, but we have the full intuitive case in mind when we are considering formalizing a mathematical intuition. The former is empirical, but is the latter?

What it seems like to me is that when we are creating new mathematics we get an intuition about what it is we are considering and work from there using our various intuitions, and build a sort of machine in our minds. It is a machine that is abstract and we know very well whether we are certain that it works in one way or another and whether one sort of fact is true about it or not, at least on a superficial level. So we take the facts that we consider to be true about this mental machine and try to encode it in formal language. Because it is an object of our intuition, there is no hidden information about it, everything that we consider to be true about it we know for certain is true about it, because we have created it.

So when we find our encoding gives us a result we did not mean it to give, do we shift our intuition or do we try to shift our axioms?

Is it the case that all naive mathematics was developed in this way, and that we do the best to preserve what aspects of the intuition of other mathematicians by internalizing proof systems along with their naive writings?

Mathematics is clearly bifurcated here, and if we consider theorem cranking, we get one aspect of the picture, but we don't get the whole picture. One side of this is the formal proof process, which draws on intuition, but is bounded by the axioms. Our intuitions are guided by the formally encoded axioms, and we attempt to encode our reasoning in to a formal proof. If we succeed then we have proven what we have intuited as correct, and if it appears that we cannot do so, we try to prove it false by encoding intuitions about why it might be false and seeing if we can make this logically coherent and complete.

So I say that a large aspect of mathematics is the process of developing a sort of mental machinery, and translating it in to a logical formalism and then using that logical formalism to guide future intuition about the mathematical structure. The other aspect is that of gaining the mathematical intuition back from the axioms and deriving new ideas and formalizing them. If you aren't the originator of the theory, then you can't make any claims about the formalism fitting the original intuition correctly, you have to fit your intuition to the stated axioms. Nonetheless, what is this intuition? Are we observing a mental process and recording it in generally agreed upon formalisms? Does this indicate any aspect of Empiricism in part of the mathematical process?

 

[Master_Ghost_Knight]

I was going o make an effort to quote you, but I decided that it just doesn't worth the effort.
If there were any doubts in my mind that you are completely oblivious to what you are talking about, you have just taken them all. I have gone into great lengths not to insult you, good for you my day isn't going so bad.
1. Reverse mathematics doesn't mean what you think it means, go back to the wiki page and see if you can find any of the properties you have listed there. You won't find them because that is not what it is about.
2. It is indeed true that you can get a different set of axioms and get something else like mathematics, but once you have done so your "new field" is completely divorced from every other and there is no interchange between them.
3. Very often I have found myself with mathematical propositions that were so obviously true (informaly) that you might as well just skip the process of proving it, just to turn out that they were fundamentally false and absurd (every mathematician who is a mathematician has gone through this experience). You do not, and cannot say that because something is obviously true to you that therefore you are given a free pass to try and back track the necessary requirements and fully integrate them in already existing work. Because what you just did is an arbitrary subjective assumption, which not only fails to have any basis to justify that leap, but also by making that assumption you made the entire work most highly probable incoherent and false. You have just killed mathematics. I don't know what you think that is, you can call it pseudo-math, a brain fart, you can call it a one legged donkey on the butt crack of a gypsy, but you must certainly must not call it math. You completely and utterly annihilated yourself with only 1 sentence, and for my disapointment all I did was to sit and watch you make it.
You are proven absurd.

This discussion is over.

 

[Zetetic]

I was going o make an effort to quote you, but I decided that it just doesn't worth the effort.
If there were any doubts in my mind that you are completely oblivious to what you are talking about, you have just taken them all. I have gone into great lengths not to insult you, good for you my day isn't going so bad.

:roll: Don't kid yourself about the possible impact the words you type on the internet can have on me or my mood. Now, we can have a good discussion, where you actually give me the benefit of the doubt (assume that what I mean to say is probably reasonable, and if you are unclear about it, ask what I really mean), and I return the favor, and we can try to figure out what each of us means like civilized adults... or we can continue in the way that you have started doing, acting like children who can't subdue their desire to ridicule when they misunderstand what they are seeing.

1. Reverse mathematics doesn't mean what you think it means, go back to the wiki page and see if you can find any of the properties you have listed there. You won't find them because that is not what it is about.

What exactly is it that you think I am saying about reverse mathematics? To be clear this was a tangential point, not saying anything directly about Reverse Mathematics, but rather about the class of ideas that makes it interesting to me:

but what makes it really interesting is the fact that we can run into theorems that are to us intuitively correct, but do not follow from the axioms we have devised. For instance, we might like that the Continuum Hypoethsis be correct, but it turns out that it is independent of the ZFC axioms, and a great deal of resources over a century had been devoted to trying to prove it to be correct or false. It was not until Dana Scott developed his forcing technique that the problem could be settled, and it was not intuitively obvious to any but a few mathematicians that an independence proof was needed until the later half of the twentieth century.

My point, if you'll allow me to try to state it more clearly, is that the very fact that we can ask which axioms are appropriate to formalize mathematics indicates that mathematics is not totally reducible to logic, although we can (or at least, we seem to be able to) formalize it using logical structures. There is some axiomatic relativism in the sense that there are multiple consistent models that satisfy the theorems of arithmetic, for instance. If one axiom system is strictly stronger than the other, how do we know for certain which one to choose? If you don't think that Reverse mathematics was developed at least in part to try to classify axiom systems by their strength relative to one another, and to devise the weakest axiom scheme allowable that can cover 'ordinary or pre-set theoretic' mathematics (an axiom scheme that is significantly weaker than ZFC), then you are wrong. If you think I was asserting something else, then you are confused.

I would not only consult the wiki page, by the way. I am referring at least partially to infomration found in "Subsystems of Second Order Arithmetic" by Stephen Simpson, which is, as far as I know, the only full length text on the subject. If you desire a copy you can procure one online in various ways or you can purchase it legitimately. I can try to quote examples from the book for discussion.

From that book, in the introduction;

"Foundations of mathematics is the study of the most basic concepts and logical
structure of mathematics, with an eye to the unity of human knowledge. Almost all
of the problems studied in this book are motivated by an overriding foundational
question: What are the appropriate axioms for mathematics? Through a series of
case studies, these axioms are examined to prove particular theorems in core mathematical
areas such as algebra, analysis, and topology, focusing on the language of
second order arithmetic, the weakest language rich enough to express and develop
the bulk of mathematics.
In many cases, if a mathematical theorem is proved from appropriately weak
set existence axioms, then the axioms will be logically equivalent to the theorem.
Furthermore, only a few specific set existence axioms arise repeatedly in this
context, which in turn correspond to classical foundational programs. This is the
theme of reverse mathematics."

My assertion is that because we have such questions about which axiom system is appropriate, that in mathematics we are doing more than what we do in logic proper, and that this is because we are encoding intuitions in to formal systems and attempting to work with them there because it is the only way that we can maintain consistency.

I claim that mathematics is applied logic, and not totally reducible to pushing logical formalisms around. It is generally accepted that Hilbert's Formalism was a deeply flawed endeavor. http://en.wikipedia.org/wiki/Foundations_of_mathematics#Foundational_crisis at the very least you have to concede that there are theorems that we can state that mean something to us that cannot be neatly packaged into an axiomatic description of what we mean.

2. It is indeed true that you can get a different set of axioms and get something else like mathematics, but once you have done so your "new field" is completely divorced from every other and there is no interchange between them.

Not what I am talking about. I assume you've studied model theory, right? You seem to make a lot of authoritative claims here so you must have some familiarity with it? So do you see what I am talking about when I am speaking about encoding something in formal language by constructing a model?

What about the attempts to formalize set theory? Frege certainly had an intuition about the objects he wanted to formalize, but his formalization was flawed. Russell's paradox showed this to be the case. His formalism was inconsistent. Logicians put a great deal of work into formalizing how one can prove that a system of axioms is sound, compact and complete. This is because there was a disconnect between our intuitions and logical formalism, is it not?

What about the creation of Measure theory to try to deal with absurdities like everywhere discontinuous functions or the Weierstrauss function? The theories of real analysis were effectively amended to better fit our intuitions about about functions, or do you disagree?
What is your take on why such a heavy duty formalism as measure theory was developed?

3. Very often I have found myself with mathematical propositions that were so obviously true (informaly) that you might as well just skip the process of proving it, just to turn out that they were fundamentally false and absurd (every mathematician who is a mathematician has gone through this experience). You do not, and cannot say that because something is obviously true to you that therefore you are given a free pass to try and back track the necessary requirements and fully integrate them in already existing work. Because what you just did is an arbitrary subjective assumption[

You misread what I said:

So I say that a large aspect of mathematics is the process of developing a sort of mental machinery, and translating it in to a logical formalism and then using that logical formalism to guide future intuition about the mathematical structure. The other aspect is that of gaining the mathematical intuition back from the axioms and deriving new ideas and formalizing them. If you aren't the originator of the theory, then you can't make any claims about the formalism fitting the original intuition correctly, you have to fit your intuition to the stated axioms. Nonetheless, what is this intuition? Are we observing a mental process and recording it in generally agreed upon formalisms? Does this indicate any aspect of Empiricism in part of the mathematical process?

I hope the part in bold clears things up for you on this point, otherwise, let me know how I can be of further assistance.

 

[Master_Ghost_Knight]

Let me quote you something.

but what makes it really interesting is the fact that we can run into theorems that are to us intuitively correct, but do not follow from the axioms we have devised

You just don't get it do you?

 

[Zetetic]

Let me quote you something.

but what makes it really interesting is the fact that we can run into theorems that are to us intuitively correct, but do not follow from the axioms we have devised

You just don't get it do you?

I'll admit that I did not state what I meant very clearly, I wrote it haphazardly and quite late in the day (yes, it was 5:00 am and I had woken up at 9:00 am the previous day, one is apt to be incoherent in his writing under those circumstances), so if you demand it, I will apologize for posting in such a state and retract that entire post. In any event, that single statement was ancillary to my overall point. There is still some ambiguity when choosing the -correct- system of axioms to satisfy pre-set theoretic mathematics (and even what we mean by correct). The people who developed the formalisms in which to embed pre-set theoretic mathematics did not have a trivial task, and we are still tweaking foundational structures.

I am not arguing about whether the logical formalism is consistent, I'm not saying that first order predicate logic is fallible or that second order logic is fallible, I am saying that mathematics is distinct from the subsystems of second order logic that are used to formalize it, and that the process of taking some intuitive concept of a mathematical structure and encoding it into a formal system is more an art than it is a straightforward exercise. So I am saying that in building mathematics, there are points at which we do this, so mathematics is not strictly reducible to logical formalisms.

So do you have anything to say about this? Why do we adjust the framework we are using when we run into problems? Why was it necessary to build measure theory and not just keep cranking out what we could in the older formalisms? Why is there debate over whether a theorem that requires the axiom of choice is acceptable or not? It is because what we mean by mathematics is not an absolute and precise thing, no matter how precise the formalisms in which we encode our ideas are. We have a distinct set of mental objects and processes we consider, and over the years there have been several different thoughts on how these ideas should be formalized.

The issue is not whether a proof is sound, it's whether the axioms were chosen carefully to represent our meaning.

You can take the position that everything deducible from a set of axioms is mathematics, but that is not generally an accepted stance. If I tell you that any syllogism is mathematics, you would look at me as though I had two heads. This is not what we mean by mathematics "Socrates is a man, all men are mortal, therefore Socrates is mortal", what we mean by mathematics are statements of a certain type! But what is that type of statement? What are the boundaries of mathematics? When have we left mathematics and gone into a different part of applied logic?

 

[Master_Ghost_Knight]

Even if I am to consider that you were completly drunk when you wrote that. You are still atributing all the propreties that are anti-thetical to mathematics.
Mathematics has no mechanism of verification by probing the real world and allot of work is heavilly interdependent and recursive, making it prety hard to track any hierachy. In order to ensure acurracy, you can not just present evidence (there is no such thing as evidence in math), you have to present proof, as in absolute proof that it is unequivocally correct (which fortunatly you can in math). Were we to permit a single unverified statment and the all field is no longer reliable, destroying mathematics and making it usefuless do to the fact of it being inevitably false. There is no room for "I feel it should be like this" or "it is intuitive to me", there is no room for opinion, it is put up or shut up.

 

[twoism]

I think philosophy is mostly a giant load of BS created by people who refuse to get real jobs.

Then you don't know philoophy. Although I give it to you that there isn't much to be done in the developmet of threads of reasoning, but anyways I still believe that if people were more philosophycal savy (which most people don't have the slightest clue of what is about much less use it) then there wouldn't be loads of the bullshit we see today. Note: topic best fit in philosophy.

It's not as ridculous as it may sound. I'm with PZ Myers on this one: modern phliosophy asks a lot of good questions - but the answers are pretty lousy on the whole. descarte did his return to i think therefore i am existing and look what conclusion he came up with from that.

 

[Master_Ghost_Knight]

It's not as ridculous as it may sound. I'm with PZ Myers on this one: modern phliosophy asks a lot of good questions - but the answers are pretty lousy on the whole. descarte did his return to i think therefore i am existing and look what conclusion he came up with from that.

I would say that is more of a case of intelectual lazyness rather than a fault of philosophy (which is plagued by the lack of formalism and objectivtity within philosophical circles).