What is the proof of 2 + 2 4

«2 plus 2 is not always equal to 4»

NZZ am Sonntag: Mr. Specker, you gave the title “The glamor and misery of logic” to a lecture at ETH Zurich. What is so shiny and what is so poor about this discipline of thought?

NZZ am Sonntag: Mr. Specker, you gave the title “The glamor and misery of logic” to a lecture at ETH Zurich. What is so shiny and what is so poor about this discipline of thought?

Ernst Specker: Without logic, today's information technology would not exist. It had to be recognized first that and how language can be formalized before it became possible to solve so many tasks with the help of computers. More important to me than the applications, however, is the influence of logic on mathematics itself.

How did this influence manifest itself?

In a first step, Gottlob Frege expanded the Aristotelian logic into so-called predicate logic. The logical principles have been formulated that allow conclusions such as the following: If cats are mammals, the head of a cat is the head of a mammal. Thanks to this new logic, the structure of mathematical proofs has been clarified. In a second step, geometry was freed from intuitive notions of space and the relationships between points and lines were interpreted in a purely abstract way. David Hilbert put it as follows in 1891: “Instead of“ points, straight lines, planes ”you have to be able to say“ tables, benches, beer mugs ”at any time. So no descriptive ideas should flow into the evidence.

What else could you achieve thanks to this new perspective?

It was applied to the theory of natural numbers. But this raised a new question: Are the postulated relationships between the numbers also free of contradictions, like those of elementary geometry? Or can it be proven that: 0 = 1? Probably even the greatest skeptics would not have dared to express such a suspicion if Bertrand Russell had not shown that precisely this is possible in the original Fregeschen system. To counter the doubts, Hilbert designed the program named after him, which was supposed to prove the consistency of number theory. Evidence was interpreted in this way as a sequence of positions in the game of chess. Starting from a starting position, called axioms, it should be shown that a certain final position cannot be reached with legitimate moves. In chess, for example, this would be a position where the white king is in the middle of all eight black pawns, in number theory the "position" 0 = 1.

Did Hilbert's program succeed?

Not as planned. In 1931 Kurt Gödel proved in his now famous “Incompleteness Theorem” that Hilbert's goal cannot be achieved with the intended elementary methods. However, the program has led to many insights that show how formal systems work. Incidentally, in the farewell lecture you quoted at the ETH, someone asked me what logic actually is.

What did you answer?

Logic is the art of speaking sensibly about things that you don't understand. There is a certain truth in that, because logic is something formal. Formalized thinking means that it is not the objects that matter, but their mutual relationships. You don't have to understand anything about number theory in order to check whether a number theoretic theorem has been correctly proven or not. This is exactly how you can tell by logical conclusions whether a Sudoku is solvable or not.

However, concretely mastering one of these number puzzles, which has come into fashion, is more fun. Do you solve Sudokus?

Preferably with a grandson. Not in competition, of course, but together. Once you have overcome a difficulty, it often runs automatically for a while. That gives a feeling of happiness.

Automatically? Not everyone feels that way.

But everyone uses logical methods in solving, perhaps subconsciously. For example, he uses the principle of exclusion. If only two numbers are allowed in a field, but one of them is not, it must be the other. The nice thing about logic, for example, is that it provides a method that is sufficient to decide whether a Sudoku works or not. The situation in number theory is more complicated. If an assumption there does not lead to a contradiction, one must not automatically assume that it can be proven. In number theory there are propositions that are neither provable nor refutable.

Does this “incompleteness” principle, already mentioned, justify Gödel's position as the greatest logician since Aristotle and Leibniz, which is sometimes euphorically granted to him?

To classify scientists or artists in this way is unreasonable in my opinion. But if big names are to be mentioned, then Gottlob Frege and David Hilbert must certainly not be missing.

But why is Godel so famous?

Fame is often based on misunderstandings. Gödel's theorem, for example, says that for all sufficiently sharp axiom systems there is a theorem that cannot be proven in them. In many representations, the limitation of formalized thinking was inadmissibly derived from this: humans simply cannot do everything. If one wants to understand scientific achievements correctly, one must not ignore the context of their validity. I explain the importance of the environment to my grandchildren with the following task: A mayor would like to travel to Rio for the carnival with his daughter, his lawyer and his wife. However, there are only three seats left on the plane. Nevertheless, the group can fly with you. How is that possible?

The daughter sits on her father's lap.

This is what my granddaughter argued, but it is not considered a valid solution. The solution expected at school is: the daughter is the lawyer's wife. As this example shows, 2 plus 2 is not always 4.

In 1949 you were a postdoc at the Institute for Advanced Study in Princeton, where Kurt Gödel and Albert Einstein had their offices. Did you meet the two of them?

We were told not to bother Einstein, but at least I heard him at a lecture. Godel also lived very withdrawn. I never saw him at the afternoon teas. Once I was invited to his house with other young mathematicians and met his lovely wife. Another time he received me, probably because of his friendship with Bernays. I wanted to show him two of my results, but he didn't respond. That disappointed me a bit.

Did you regret never having been able to discuss logic with this strange person?

No. I hated the people who tried to get close to Einstein or Gödel. There are people who are recognized as great minds, but who then suddenly appear to be quite naive. Gödel thought, for example, that fundamental questions of the theory of infinite sets can be decided on the basis of logical principles.

So for you, logic was never the only access to truth?

I always saw myself as a mathematician and practiced logic like other areas of mathematics. There are also different logics in which you can move like a chess player solving chess problems. Questions raised by logic interested me, but why should logic alone solve the problems of truth? That can't work. Man is much more than he knows about himself.

Isn't that why mathematicians are unsettled even if logicians were to prove that their building was built on sand, as one of Hilbert's students put it?

For a short time at most. People quickly get used to initially unsettling things. As a whole, however, mathematics is arguably the best-founded human enterprise. In my opinion, theories that presuppose the concept of infinity are built on sand. For example, it is not understood why physics apparently cannot do without real numbers. We don't even really know what "number" means, but it shouldn't be said out loud.

At the beginning of the last century, questions about the fundamentals of mathematics were still the focus of broad interest. In 1987, after you retired, ETH Zurich did not even fill your chair in logic.

Which brings us to the second part of your initial question. Why did I speak of the misery of logic in my lecture? In the Middle Ages philosophy was considered a maid by theologians. In continental Europe, in contrast to the Anglo-Saxon cultural sphere, logicians are now dependent on the understanding of mathematicians when it comes to filling chairs. And it is apparently not enough to point out that modern computer technology is unthinkable without the preparatory work of the logicians and that much can still be expected of logic in the long term. The maid has done her job, the maid can go. But that doesn't make me too sad. What scientist other than a logician can say truer of himself than the following sentence: "There are three kinds of logicians, those who can count on three and those who cannot"?

Interview: André Behr

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