"atyy said: ↑
In fact, Bourbaki also says that the language of mathematics rests on the informal language of physics, biology and psychology. For example, they say in the Introduction to their Theory of Sets that one needs to assume that we know what is meant by a letter of the algebra being "the same" in two different places on a page.
In the same Introduction they also say "The verification of a formalized text is a more or less mechanical process". Again that is physics, implicit in the word "mechanical".
They also say that it is impractical to carry out all mathematics in the formalized way, and they will therefore use informal arguments in which the existence of the intuitive natural numbers will be assumed before any formal arithemetic is defined.
As far as I can tell, my views are very Bourbakist 
"
lavinia said:
In practice mathematics is done through intuition and insight. Formalism is always an after thought - part of the process of verification - but not the source of ideas.
I am a firm believer in the above statement from lavinia being true.
Math is important, but I once worked out an observation about intuition, oft called insight, being foremost in the ultimate gain of human knowledge. I call it the Race Team principle.
Suppose we observe a successful, typical race team that races cars, a NASCAR team for example. The "win" seems to depend upon the driver having an intuition that most closely approximates real physics. In a nutshell, he, or she, must quickly calculate the best balance between tire adhesion and centrifugal forces. This seems to me to be a rather pure example of intuition. On the other hand, the car cannot win without the skills of a top notch mechanic, no matter the naturally gifted extent of the drivers intuition.
The mechanic uses skills of physics that can be taught, thermodynamics, material selection, a variety of tools which he, or she, knows how to apply quite well... even if it involves some head scratching on occasion. This is not unlike a well trained mathematician and his, or her, tools. The driver wins because he, or she, is an exceptionally well tuned child to Mother Natures laws of motion and friction. So well tuned that the driver intuitively knows where to go when there is no time allocated for head scratching; a quick conjecture in the raw... a sudden eureka of sorts.
This is not to say that the best driver is not an accomplished mechanic, nor the best mechanic an accomplished driver, and the best of both would therefore be a driver/mechanic that surpassed any of either. But in reality almost all gain is still made by teams. And so is it true of the giants of physics whom at least stand on the shoulders of their team-mates.
As examples, many of our scientific giants, our scientific "drivers", could apparently see, could conjecture, Nature's geometry before hashing out the mathematical proof. Copernicus (Heliocentricity), Kepler (eliptical orbit), Newton (the most far-flung falling cannonball), Einstein (rods do get shorter, Equivalence), Feynman (his diagrams, the
Lost[/PLAIN] Lecture) and more.
Back to subject, I found a purported partial english copy of Principia Mathematica here: http://www.olimon.org/uan/principia_3.pdf .
By substituting 1 and 2 in the above address, I find earlier sections of the book. However, by substituting 4, I find no more. Is the above address 3, the final end of it all? The last page of the series looks incomplete. A glance at
http://www.olimon.org home reveals the main website is in Spanish.
Wes
...
