Master_Ghost_Knight said: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.
Master_Ghost_Knight said: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.
Master_Ghost_Knight said: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.
Master_Ghost_Knight said:, 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.
Master_Ghost_Knight said:... 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.
Master_Ghost_Knight said: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.
Master_Ghost_Knight said: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.