Mathematics and Scientific Laws

1. ______ Despite living 250 years before the invention of the computer program, Leibniz came very close to the modern idea of algorithmic information. He had all the key elements. He knew that everything can be represented by binary information, he built one of the first calculating machines, he represented the power of computation, and he discussed complexity and randomness.

2 .______ If Leibniz had put it all together, he might have questioned one of the key pillars of his philosophy, namely, the principle of sufficient reason – that everything happens for a reason. Furthermore, if something is true, it must be true for a reason. That may be hard to believe sometimes in the chaos        Gottfried Leibniz           and confusion of everyday life, in the contingent ebb and flow of human history. But even if we cannot always see the reason (perhaps because the chain of reasoning is long and subtle), Leibniz asserted, God can see the reason. It is there. In that he agrees with ancient Greeks, who had originated the idea.

3. ______  Mathematicians certainly believe in reason and in Leibniz’s principle of sufficient reason, because they always try to prove everything. No matter how much evidence there is for a theory, such as million of demonstrated examples, mathematicians demand a proof of a general case. And here is where the concept of algorithmic information can make its surprising contribution to the philosophical discussion of the origins of limits of knowledge. It reveals that certain mathematical facts are true for no reason, the discovery that flies in the face of the principle of sufficient reason.

4. _______ Indeed, it turns out an infinite number of mathematical facts are irreducible, which means no theory can explain why they are true. These facts are not just computationally irreducible, they are logically irreducible. The only way to prove these facts is to

assume them directly as new axioms, without using any reasoning at all. The concept of an “axiom” is closely related to the idea of logical irreducibility. Axioms are mathematical facts that we take as self-evident and do no try to prove them from simpler principles. All formal mathematical theories start with axioms and then deduce the consequences of these axioms, which are called theorems. That is how Euclid did things in Alexandria two millennia ago, and

his treatise on geometry is the classical model for mathematical exposition.

5. ______ In ancient Greece, if you wanted to convince your fellow citizens to vote with you on some issue, you had to reason with them – which probably how the Greek came up with idea that in mathematics you have to prove things rather than discover them experimentally. In contrast, previous cultures in Mesopotamia and Egypt apparently relied on experiment. Using reason has certainly been extremely fruitful approach, leading to modern mathematics and

   Alan M. Turing                     mathematical physics and all that goes with them, including the technology for building that highly logical and mathematical machine, the computer. Actually this approach that science and mathematics have been using for more than two millennia seems to crash now.

6. ______  In a 1936 issue of the Proceedings of the London Mathematical Society, Alan M. Turing began the computer age by presenting a mathematical model of a simple, general-purpose, programmable digital computer. He then asked: ”Can we determine whether or not a computer program will ever halt?” This is Turing’s famous halting problem. Of course, by running a program you can eventually discover that it halts, if it halts. The problem, and it is an extremely fundamental one, is to decide when to give up on the program that does not halt. A great many special cases can be solved, but Turing showed that a general solution is impossible. No algorithm, no mathematical theory, can ever tell us which programs will halt and which will not.

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