[Lloyd Gillespie]

Hi Guille, this is an excellent thread starter, as it hits the point I have been recently mulling over in my head. Your point, "Probably you know that the basis of mathematics is logic. But, how exactly does mathematics relate to logic? Is it logic's language, or just a type of use of it?", is exactly what I will try to address.

If we truly look at the mind and its possessions, we find the logic processor you mention, and we also find the mathematics processor___why are these two processors so different? IMO, they are drastically different, as are logic and rationality, as you mentioned. Now, what makes these differences? The major difference I see is logic has the intuitive ability to use the infinite probabilities processor, and mathematics is bound to absolute science knowledge and facts, i.e., the largest infinite number mathematics can process is a huge rationalized infinity, or a correspondence logic of infinity to finiteness, which is mainly the same but different[isomorphic] thing, and the mathematics of statistical mechanics probabilities based on incomplete ergodic theorems and axioms, which work for larger than true ground micro states, and the less than true absolute macro states of phase spaces. Thus, we can use mathematics only to a certain level of certainties, limited by the impossible penetration of absolute micro and macro phase space states___while logic, on the other hand, can use its intuitive rationality, mathematics and probabilities of absolute probabilities. Now, the question becomes, how can we process our mind's ability to do logical absolute probabilities, mathematically___or can we at all?

The way I see it, the mind possesses the infinite intuitive process of absolute probability math, yet there is no scientific math, yet, possible to represent this fact. Now, a few examples may make this clear, such as, our mathematical ability to represent quantum packets, yet no true math to represent the actual position and momentum of the electron. Or, our mind's ability to know infinity as a real entity, either through the non-ending rational facts of math, or the facts we witness by experiments and just looking up at the sky at night. We see finite matter and infinite space. We also realize we possess an infinite imagination, yet, we can not mathematize it. So, there are several walls of impenetration between logic's and math's abilities. Kurt Godel was the most advanced in this investigation, when he died. He mentioned the trans-finite rationality, which is very interesting, as it seems a long way past the reasoning powers of Witgenstein and Russell. At least Godel pointed us in the direction math must take to solve the world's and universe's many incomplete problems. He also, along with Brouwer, was the greatest mind in working on the expansion of intuitionistic logic, the field I am always most interested in, as a possible way to achieve the absolute probability math, really a new whole math, which I believe must be based on new and true interpretations of one, zero and infinity. I think if we could develop such a new whole math, we may be able to develop the complete link between logic and math, as Russell, Whitehead, and many others tried, yet failed. Godel did not think his incompleteness theorem spelled the death of ever figuring out the advancement of mathematics, and neither do I, but I do think it will take a new whole mathematics to truly join logic and math completely, and only then will we be able to truly answer and prove the toe, logically and mathematically sound.

IMO, the way we process infinity___logically, scientifically and mathematically___is the key...

Regards,
Lloyd

Originally Posted by Guille

As a first thread to this forum, I want all of us to keep the high level of abstraction that Robert asks in his introductory thread. I believe that the theme I have chosen is perfect to begin an overal view on the nature of mathematics.

Probably you know that the basis of mathematics is logic. But, how exactly does mathematics relate to logic? Is it logic's language, or just a type of use of it?

Well, In my opinion, mathematics is the language that humans use in doing science to study nature. And all of science has a core which is pure logic. But science is the study of nature. Not the study of logic or the study of the relationship between mathematics and logic. This arises a question in my mind: If the logicists of the 20th century were correct when they said that humanity was (and still is) going through a des-philosophicalization and intro-scientification, then, will there be a science one day that studies mathematics itself, and the relation of mathematics to logic, or of mathematics to science? In theory, the answer should be yes, but this doesn't make sense to our minds. It dosn't make sense because the actual fact of questioning and considering the relationship between the study of nature (science) and the language of the study of nature (mathematics) is itself philosophical. Now, coming back to the top of tis paragraph, well, mathematics is thus a kind of logic, or beter said, a way of using logic. So mathematics is not the language of logic. Mathematics is the study of mathematical logic, and the language of mathematical logic, but not of logic itself. A way of understanding the idfference between mathematical logic and logic itself is by studying human beings: each has a logic, that has been created in the mind by their experiences in their lifes. But these logics differ: differ first inside the same human, who can change of logic by a stron expirience, differs between humans, and, of course, differes to mathematical logic. But it's still logic. Now that we have clear that there are different kinds of logics, a question arises: Are there independent languages for each kind or type of logic, or can a language be created to represent the LOGIC itself, of any kind? The answer depends absolutelly in one property: if the types of logics all have a connection of the type that they can be different but still proccessed in the same way, then they can share a language, but if they don't have a connection of the type that they can be different but still proccessed in the same way, then there can be no language for all, because they are porcessed in different ways. I cannot determine the correct answer myself. But I believe it's more possible that no lanugage exists for all the logics, because we, humans, have invented a laungage for mathematical logic, but we don't manage to invent a language for the logic of our minds. We still odn't understand how it works, and this is because we study mind logic by mathematical logic, but, if mind logic can't be related with mathematical logic, then it must happen that they are processed in different ways, and, thus, that there can be no shared language.

I will continue to expose my porfound thoughts in this thread now, I give you all the opportunity to consider my thoughts and discuss them, and expose yours.