[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.

[Guille]

Now that we have clarified the relationships between mathematics logic and science, and we have adetermined that mathematics is not the language f logic itself but a kind of logic, we can proceed to describing what the relatioships between mathematics and logic are. Well, the branche of matheamtics that studies logic, which is wrongly called logic, has tried to make a logical foundation of mathematics by not using mathematics but describing it in matheatical forms. And this is, in my opinion, a mistake. A mistake, because by describing with mathematical language the mathematical study of logic, we are making logic itself mathematical, and thus, it makes logic go further and further away form nature. For example, we say that 1+1=2 which in number theory is ok, but transforme dinto logic, it's wrong: because 2 is the set of all the numbers between 0 and 1.9999... not just of 1 and 1. But of course, the logicla connection "and" is very different from the mathematical sing "+". Another mayor error is that whiles in number theory 2 is a complete new number separate from the 1s, well, in logic, 2 would not be a "new" set in that sense:: it woudl be a new name, but it would just be the adition of the previosu sets, the group of 1 and the other 1 is set 2, so set 2 is not really new.

This, we can conclude that mathematics is not equal to or good enough to describe logic. Morever, we can say that mathematical logic is not completely logical, which is a very basic principle into starting to understand the differences between "logic itself" and mathematical logic.

[AntonioLao]

We can also start by differentiating the most important distinction between logic and mathematics.

Logic is a science - it is based on the scientific method - a principles and procedures for the systematic pursuit of knowledge involving the recognition and formulation of a problem, the collection of data through observation and experiment, and the formulation and testing of hypotheses

Math is an art - it is based on the mind prior to reasoning. This is why there can never be any agreement between the intuitionists, the logicists, and the formalists. The existence of these schools of mathematical minds is the same as the existence of the trichotomy of yes, no, and maybe.

[Guille]

Antonio,

how on earth can mathematics be a prior if it has no a prior existence! It is only rationality, it has no perceptual existence.

[AntonioLao]

Originally Posted by GUILLE how on earth can mathematics be a prior if it has no a prior existence! It is only rationality, it has no perceptual existence.

As some intuitionists would say math is prior within the mind (idealized existence) before rationality (reasoning and thinking). No one has to be taught to recognize what is beautiful and symmetrical.

[Guille]

Originally Posted by AntonioLao As some intuitionists would say math is prior within the mind (idealized existence) before rationality (reasoning and thinking). No one has to be taught to recognize what is beautiful and symmetrical.

That's true. But in my opinion, logicists and intuitionists don't disagree, they just look at different aspects of maths. Intuitionists look at the prior knowledge and logicists look at base-knowledge.

What is formalism all about?

[Graybeard]

Antonio & Guille

No one has to be taught to recognize what is beautiful and symmetrical.

I think this is an aquired trait, learnt with your mother's milk. You do have to be 'taught'. I don't think we are born to recognise beauty .... if so ... we would all agree on what is beautiful and symmetrical. And that we don't do.

'Home sweet Home' has the same meaning all over the world, but each individual has a unique picture of 'Home'

Therefore the recognition of 'beauty & symmetry' is a philosophical science, ie: it can never be conclusively resolved.

If my point is valid, then I don't see how 'Math is an art - it is based on the mind prior to reasoning'

do you agree ???

Greg

[Guille]

Greg,

I'm curiose, what do you mean when you write "philosophical science"? Is the word science here just standing for the word knowledge, or you trully mean a sort of study of nature which is due to it's form of being can be called "philosphical science? If so, how is it?

[Guille]

Greg,

You have given an important point in remembering us that beauty is subjective. But this leads to several philsophical problems. First, we must define beauty. Beauty is symetry, order, proportionality, equallity, pleasurable to be consciouss about, and, most importantly (for our particular discussion), it is a concept. And as I described in the thread "The incompleteness theorem", concepts can not be shared by two different minds because the fact that they are different minds impplies that they are subjective, and thus, that they don't have the same information for none of their components or fields. So, this is why each of us have our preferences on people for their physical properties. But the physical property of beauty is an independent entity from each mind. When something is beautifull it is beautifull. Of course, each mind has preferences, and might not say it's beautifull. For example, for a mathematician, euler's identity e^iπ+1=0, is beautifull, and the new "armani collection" for fall/winter 2005/2006 is just a lot of clothes. And, for an stylist, it would be vice versa. Who is correct? None and both. Both are beautifull, the equations and the clothes. This is because independently of human minds, they havethe properties requiered for beauty previouslly described.

[AntonioLao]

If I consider myself to be an intuitionist then Guille would be the logicist and Greg would be the formalist.