[Graybeard]

Guille .... your right of course ... i should have used the word philosophy .. i don't consider philosophy a true science .... ie: as in a verifiable and testable science ..

as you say i simply used it in place of knowledge ... many people, possibly rightly do consider it a science ... would you say Democritus was a philosopher or a scientist ?

he did name the atom !!

[Graybeard]

Antonio ... thats cool ..... i have read many of your 'mot justes' and i find them very clever...

however, neither of you answered whether we are born with appreciation of beauty and symmetry or whether it is aquired?

[AntonioLao]

Beauty is in the eye of the beholder.Meaning: You use this proverb to say that different people have different opinions about what is good/beautiful/valuable. Example: Most people here hate this food, but beauty is in the eye of the beholder. People in many other countries love it and eat it every day. Like beauty, truth is also in the eye of the beholder. It is through accepted conventions by teaching and learning that people acquire different perspectives and beliefs.

[AntonioLao]

Originally Posted by GUILLE What is formalism all about?

The following were edited from the website at
http://en.wikipedia.org/wiki/Philosophy_of_mathematics
Formalism holds that math statements may be thought of as statements about the consequences of certain word manipulation rules. For example, in the "game" of Euclidean geometry (which is seen as consisting of some words called "axioms", and some "rules of inference" to generate new words from given ones), one can prove that the Pythagorean theorem holds (that is, you can generate the word corresponding to the Pythagorean theorem). According to some versions of formalism, the subject matter of mathematics is then literally the written words or symbols themselves. Then any game is equally good, and one can only play the games, not prove things about them. Unfortunately, this does not solve the epistemic problems (What are symbols? Do they exist in an eternal, unchanging realm?), does not explain the usefulness of mathematics, and renders mathematics an utterly spurious activity. A second version of formalism is often known as deductivism. In deductivism, the Pythagorean theorem is not an absolute truth, but a relative one: if you assign meaning to the words in such a way that the rules of the game become true (ie, true statements are assigned to the axioms and the rules of inference are truth-preserving), then you have to accept the theorem, or, rather, the interpretation you have given it must be a true statement. The same is held to be true for all other mathematical statements. Thus, formalism need not mean that mathematics is nothing more than a meaningless symbolic game. It is usually hoped that there exists some interpretation in which the rules of the game hold. But it does allow the working mathematician to continue in his work and leave such problems to the philosopher or scientist. Many formalists would say that in practice the axiom systems to be studied will be suggested by the demands of science or other areas of mathematics. A major early proponent of formalism was David Hilbert, whose goal (Hilbert's program) was a complete and consistent axiomatization of all of mathematics. ("Consistent" here means that no contradictions can be derived from the system.) Hilbert aimed to show the consistency of mathematical systems from the assumption that the "finitary arithmetic" (a subsystem of the usual arithmetic of the positive integers, chosen to be philosophically uncontroversial) was consistent. Hilbert's program was dealt a fatal blow by the second of Gödel's incompleteness theorems, which states that sufficiently expressive consistent axiom systems can never prove their own consistency. Since any such axiom system would contain the finitary arithmetic as a subsystem, Gödel's theorem implied that it would be impossible to prove the system's consistency relative to that (since it would then prove its own consistency, which Gödel had shown was impossible). Hilbert was initially a deductivist, but, as may be clear from above, he considered certain metamathematical methods to yield intrinsically meaningful results and was a realist with respect to the finitary arithmetic. Later, he held the opinion that there was no other meaningful mathematics whatsoever, regardless of interpretation. Modern formalists, such as Rudolf Carnap, Alfred Tarski and Haskell Curry, considered mathematics to be the investigation of formal axiom systems. Mathematical logicians study formal systems but are just as often realists as they are formalists. Formalists are usually very tolerant and inviting to new approaches to logic, non-standard number systems, new set theories etc. The more games we study, the better. However, in all three of these examples, motivation is drawn from existing mathematical or philosophical concerns. The "games" are never arbitrarily chosen. The main problem with formalism is that the actual mathematical ideas that occupy mathematicians are far removed from the minute word manipulation games mentioned above. While published proofs (if correct) could in principle be formulated in terms of these games, the rules are certainly not substantial to the initial creation of those proofs. Formalism is also silent to the question of which axiom systems ought to be studied.

[Guille]

Originally Posted by Graybeard Guille .... your right of course ... i should have used the word philosophy .. i don't consider philosophy a true science .... ie: as in a verifiable and testable science .. as you say i simply used it in place of knowledge ... many people, possibly rightly do consider it a science ... would you say Democritus was a philosopher or a scientist ? he did name the atom !!

Independently of what pro-scientists such as Dave and you say, philsophy IS testable and provable and reliable. The fact that it doesn't have these properties for science and science itself, doesn't impply it doesn't have the properties from humanity or other views. This is just like when someone things that I'm stupid and another person thinks I'm smart. What am I? Both? None? One of them? Well, I'm stupid for someone and smart for another one, that's the definition.

About democritus, true, he did give the name atom which means in greek indivisible, but, his atom was just a concept, it just meant "the smallest quantity possible" of what according to hime verything was made of. Actually his atom wasn't onyl for matter but also for space, for energy, for forces and everything else too. And anyway, because atom is just a concept originally then the name could be given to it to many many different hings, which were atoms in their time. What we call atoms are NOT atoms conceptually. Not only they are made of particles but the particles of which it's made are made of other even smaller particles. And even apart from all of this, we can say that he was both, can't we? Or wasn't aristotle both? And Leibniz? And Galileo? More recently, it is true that humans have spetialyzed more and more. Bu still, for example, we don't know were the line from philsophy to sceince goes in studying the mind/brain (cognitive sciences, as it is called).

[Graybeard]

Guille ... again of course, you are right .... the first Philosophers were the first Scientists.. Looked at in this light Science is just a specialist field within Philosophy ... it gives us the reasons.... but not the root causes .... is doesn't explain life itself .. (my own opinion is that life is the result of a chance coincidence .... given 15 billion years every coincidence or chance will come up no matter how improbable ... if successful it continues on)

About Democritus, I don't know that his atom was just a concept ....
Some of his quotes are very, very insightful ..

Here are two that i find deeply disturbing, how could he know this, given the age he was in......

Democritus...2400 years ago
"Nothing exists except atoms and space, everything else is opinion" ....
"Everything existing in the Universe is the fruit of chance & necessity"

Do you see any shades of Quantum Theory here ??

At this time Zeus was responsible for Lightning and Thunder, Poseidon for tidal waves, and Ceres for feast and famine... how did he divorce himself from the beliefs of his time?

Greg

[Graybeard]

Antonio .... that was heavy going ... but i get your point on formalism ...

The main problem with formalism is that the actual mathematical ideas that occupy mathematicians are far removed from the minute word manipulation games mentioned above.

However .. on the up side ...

Formalists are usually very tolerant and inviting to new approaches to logic, non-standard number systems, new set theories etc.

Recently, in one of your posts you asked if infinity was a prime and could 1 be added to it.....this caused me enormous headaches ... I loaded the first 500 primes into a spreadsheet and spent the night trying to find a pattern in their modulous (modulo ??)

I thought i could do with the $1,000,000 as well as anybody else .... Did i get there?? ...Well i don't drive my Porsche on the bitumen yet, i am still driving it on my playstation ... (LOL)

Greg

[Guille]

Originally Posted by Graybeard Guille ... again of course, you are right .... the first Philosophers were the first Scientists.. Looked at in this light Science is just a specialist field within Philosophy ... it gives us the reasons.... but not the root causes .... is doesn't explain life itself .. (my own opinion is that life is the result of a chance coincidence .... given 15 billion years every coincidence or chance will come up no matter how improbable ... if successful it continues on) About Democritus, I don't know that his atom was just a concept .... Some of his quotes are very, very insightful .. Here are two that i find deeply disturbing, how could he know this, given the age he was in...... Democritus...2400 years ago "Nothing exists except atoms and space, everything else is opinion" .... "Everything existing in the Universe is the fruit of chance & necessity" Do you see any shades of Quantum Theory here ?? At this time Zeus was responsible for Lightning and Thunder, Poseidon for tidal waves, and Ceres for feast and famine... how did he divorce himself from the beliefs of his time? Greg

But it's science itself which doesn't explain life, it explains parts, crust parts, but philsophy is what gives the true expelanation of things itselfs, the core.

And actually the fact that Deocritus said those things doesn't mean anything new. The mental abilities and brain capacities where equal for thos men in ancient greek and the contemporary thinkers. The difference is just the amount of background knowledge. But developing principles has no variation with the knowledge, it's the start of knowledges it self, so it just deals with abilities.

[Graybeard]

Guille ... hows this

Looked at in this light Science is just a specialist field within Philosophy ... it [Science]gives us the reasons.... [Science does not]give the root causes .... [Science] doesn't explain life itself

I meant what you meant ....

The mental abilities and brain capacities where equal for thos men in ancient greek and the contemporary thinkers. The difference is just the amount of background knowledge

I don't dispute their mental abilities ... Democritus made a gigantic leap forward in culture going from Zeus & Poisedon to the Quantum... that was my point.

We all start off on the ladder somewhere and as we can neither see the bottom or the top of the ladder the only relative comparisons are how far you climb in your lifetime ... Democritus, Newton, Maxwell, Bohr, Einstein ..... These people climbed much, much higher than i ever will.

Antonio suggested that we are born with ......

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.

But I, and you also i believe, think that ...

The difference is just the amount of background knowledge

Greg

[Graybeard]

Guille ....

Quote Guille in thread starter...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?

Having re-read the thread starter, would you consider this......

Homo-Sapiens has been around for somewhere between 30,000 to 60,000 years and in all that time our DNA has not changed. (actually 60,000 is less than an eye lid flutter in terms of evolution.

Logical deduction from this means that cave-man with a stone axe ... given the right background knowledge could arrive at the Theory of Relativity, or Quantum Mechanics, or String Theory ... with exactly the same understanding as anyone alive today .. or he could be a Docter, or Philospher.... or anything he chose to be ..

I believe that he possess's LOGIC .... (Antonios intuition ?)... over many thousands of years I believe he developed a language for this LOGIC.... he called it PHILOSOPHY....but it did not explain the lo-level root causes... within this Philosophy... subsets were developed ... Mathmatics, Science, Art etc... what do you think??

Greg