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Reductionism

Physics and mathematics deal only with abstract structure. They have purged themselves of anything with an intrinsic nature. It is not natural to think this way. It is a conceptual leap. I will explain some of what led me to this conclusion as an undergraduate.

Computers were a comparative novelty in the late 60's and I was able to work with one of these extraordinary machines. I could program it to do complex tasks using simple instructions. The low level or ``assembly language'' for computers contains instructions like move the value stored in one place to a different place or add the values stored at two locations together and put the result in a third place.

The computer itself was constructed from simple operations. You could build all the machinery that controlled the computer from three components. These are devices that have input signals in the form of voltage levels and generate similar output levels. One range of voltages corresponds to the number one or switched on and another range corresponds to zero or switched off. Devices that have inputs and output signals like this are called logic circuits.

Two of these circuits had two inputs and one output. The other circuit has a single input and output. The first circuit is a logical AND. Its output is one if and only if both of its inputs are one. The next is an OR circuit. It outputs a one if either or both of its inputs are one. The third NOT circuit has an output that is the opposite of the input. The input/output function of these three circuits is shown in Table 5.1 on page [*] with `true' and `false' substituting for 1 and 0. Every logic circuit in any computer no matter how complex can be built out of these three simple circuits.

I was struck by what can be constructed from such simple building blocks. My interest and wonder was further aroused by the idea of a Universal Turing Machine. This is a very simple computer that can simulate any program that any computer could execute. One way to characterize computer programs is with the mathematical functions they can compute. A Universal Turing Machine can compute any mathematical function that any computer can compute.

A mathematical function is a formula or procedure for uniquely defining a function output number from a function input number. Many calculators have a square root function key. Enter any positive number and touch that key and the square root of the number is displayed. Square root is a function on the positive numbers. Any formula that defines a unique output number for every input number in some collection of possible inputs (like the positive numbers) defines a function. In the 1930's there were a number of proposals to characterize those functions that could be calculated by following exact mechanistic rules. All of the major contenders turned out to be equivalent. One of these was a Universal Turing Machine. At the time Alonzo Church proposed what has come to be called Church's Thesis. This states that the functions computable by a Universal Turing Machine are all the functions that can be computed by any precise mechanistic process. Church's thesis is almost universally accepted.

The AND, OR and NOT circuits of a computer are so simple that they have little content. The important thing is how they are put together to form complex circuits. The programs that control computers are made up of very simple instructions. The computer and its programs can be completely understood in reductionism terms. There is nothing to the simple instructions a program is built from or the simple circuits used to build a computer. It is only the structure of how these simple components are assembled that has significant content.

The sense that everything is structure was expanded when I studied set theory. All of mathematics was constructed with the single primitive entity of the empty set and the single relationship of set membership. This was much like the primitive circuits that made up a computer. All of this led me to ask what is it that is structured in reality?


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