Foundation

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Lecture 1

Has computation changed anything about mathematics? In other words, has it reorganized the field of mathematics? If it makes mathematical operations easier to resolve, is that the same as changing or transforming the sense of mathematics? Let me put it this way: what if, as I showed you today, you could add the digits of double, triple, etc. numbers? Such that we could always reduce 15 to 6 in certain cases or 49 to 13 to 4? Obviously it would change out concept of addition, which usually includes, perhaps, the idea of combining discrete quantities. To put it another way, mathematics is about being able to do things with numbers and the question is, what does it take to invent a new use?

Another question I have is why does it seem that mathematics is always beyond doubt? Would it be possible to have developments in mathematics without this? Or rather: what enables mathematics to develop? How did Descartes invent the concept of a coordinate system? How did he think of combining algebra with geometry (previously two distinct fields of mathematics)?

Someone once said (I think it was Wittgenstein) that to solve a mathematical problem even God would have to use mathematics.

This statement, perhaps more than any other, sums up the problems of this seminar.

Only that here, the seminar is looking at computation.

What I am asking is in what way has it changed architecture? But maybe this is the wrong kind of question because obviously it has. Maybe it is more pertinent to ask in what way it has changed architecture’s relation to mathematics, and specifically, but not exclusively, geometry.

Geometry is an obvious foundation in architecture. By this I mean we simply can’t imagine what it would be like to develop a design without it. But this is not yet to say in what sense it is internal to architecture. In some cases it is simply instrumental – tells us in specific terms what the dimension of such and such a space needs to be in order to accommodate such and such activity. In other cases it informs us of principles of design, provides us a model of meaning. But conflicts emerge. And here it is important to note that the meaning of geometry as a “model of meaning” changes.

Has there ever been a theoretical argument about architecture that didn’t somehow privilege geometry as a model of meaning? Could we imagine the geometrical properties of shape, form, scale, proportion, etc., as somehow completely arbitrary, like the number of inches in a foot? Doesn’t this mean in some sense we take geometry to be highly relevant? And not just because it is consistent and doesn’t change its laws from one day to the next, but because it seems to point to a transcendent reality, is somehow always true?

An equation is either true or false. There isn’t any space between in which it is sometimes true and sometimes false. It is like a piece of logic: all mammals are warm blooded, dogs are warm blooded, therefore dogs are mammals. An algorithm on the other hand is merely a list of instructions. And here it isn’t possible to say whether or not a list of instructions is true, is true compared to some reality, is universally true. It is merely an operation. And it is this contrast that is worth considering, insofar as the algorithm is the basis of computation, and computation is the basis of newly shifting landscape of geometry as a model of meaning.

What is a model of meaning?