We have discovered ways to automate some discoveries. But are our axioms necessarily the result of the evolution of human thought anyway?

Question: What role does human creativity play in our understanding of formal scientific truths?

Stephen Wolfram: We routinely go out into the computational universe to find algorithms which when we as humans look at them we say that is really clever, that is you know it’s a really neat thing and it’s something where if there had been a human creating them we would have been very impressed by that human’s creativity, but actually these things were just found by searching this sort of formal abstract universe of possible programs. 

So I think it’s sort of an interesting question when we look at different areas of sort of human endeavor to what extent what we have found, what we create, is something that is a feature of kind of us as humans and to what it sort of... to what extent it sort of... what is necessarily out there. 

One area I’ve thought about quite a bit is mathematics and the question is, is the mathematics that we have today sort of a necessary kind of formal structure, or is it something that really is more a reflection of the particular history of human mathematics. And so one thing that we can think about is this: if we look at sort of all the mathematics that has been done—mathematics is a field of inquiry where on thinks one is starting from a collection of axioms and then deriving all these theorems of what is true about mathematics.  The complete axioms for all the mathematics that has been done in the last however many years fit on a page or two of something therein the "New Kind of Science" book for example actually displayed on a couple of pages. So that is sort of the raw material for all of mathematics is a couple of pages of axioms.  From those axioms about three million or so theorems have been derived in the history of mathematics. And the question though is: why those axioms?  Why not other axioms? 

Well we can sort of think about the universe of all possible axioms.  We can just imagine kind of enumerating possible axiom systems that one can formally consider and we do that and we can say out of this universe of all possible axiom systems where do the axioms that correspond to our particular mathematics lie? And so I know the answer to that for things like logic for example.  I know that logic, if you were to enumerate all possible axiom systems, logic is about 50,000th axiom system that you’d find in that enumeration, so realizing that it makes one think about sort of why this mathematics and not some other? And what realizes is really the mathematics we have today is something that is a direct historical consequence of ideas that existed in ancient Babylon, arithmetic and geometry and so on, that got sort of generalized to give us the mathematics we have today and that is the mathematics that we use to kind of make our descriptions in physics and do our engineering and so on. 

What one realizes is that there is this whole sort of universe of other possible mathematics that is out there, that in fact in many cases can be much more powerful in describing things that we see in the natural world and so on, and that form sort of the basis for a lot of new directions in science and technology and elsewhere. So in a sense the mathematics that we have today is very much it’s a great historical artifact.  It’s probably, if one looks at the history of civilization, mathematics as it exists today is probably the single largest artifact in the sense that more hands have been involved in sort of molding the particular intellectual thing that has been created than anything else in history, but we have to realize that it is very much a human artifact created from its history, not something that is sort of a necessary feature of the way that sort of formal systems of the universe work. 

Recorded July 26, 2010
Interviewed by Max Miller

Image courtesy Flickr user Creativity103.