Theorem Proof Quotes

[...] provability is a weaker notion than truth

There was a seminar for advanced students in Zürich that I was teaching and von Neumann was in the class. I came to a certain theorem, and I said it is not proved and it may be difficult. Von Neumann didn’t say anything but after five minutes he raised his hand. When I called on him he went to the blackboard and proceeded to write down the proof. After that I was afraid of von Neumann.

We often hear that mathematics consists mainly of 'proving theorems.' Is a writer's job mainly that of 'writing sentences?

Occasionally, I get a letter from someone who is in “contact” with extraterrestrials. I am invited to “ask them anything.” And so over the years I’ve prepared a little list of questions. The extraterrestrials are very advanced, remember. So I ask things like, “Please provide a short proof of Fermat’s Last Theorem.” Or the Goldbach Conjecture. And then I have to explain what these are, because extraterrestrials will not call it Fermat’s Last Theorem. So I write out the simple equation with the exponents. I never get an answer. On the other hand, if I ask something like “Should we be good?” I almost always get an answer.

The theorem can be likened to a pearl, and the method of proof to an oyster. The pearl is prized for its luster and simplicity; the oyster is a complex living beast whose innards give rise to this mysteriously simple gem.

She wanted to tell him so much, on the tarmac, the day he left. The world is run by brutal men and the surest proof is their armies. If they ask you to stand still, you should dance. If they ask you to burn the flag, wave it. If they ask you to murder, re-create. Theorem, anti-theorem, corollary, anti-corollary. Underline it twice. It’s all there in the numbers. Listen to your mother. Listen to me, Joshua. Look me in the eyes. I have something to tell you.

Scientific proof is inevitably fickle and shoddy. On the other hand mathematical proof is absolute and devoid of doubt.

That story is proof of the theorem that then as today in Chicago, the mysterious equation of whiskey plus music equals what can only be called happiness.

But you can't prove God exists. And isn't that what all science is ultimately about? Proving theories about the universe?" "Provability is not truth, Caro. Godel's incompleteness theorem tells us that, if we didn't already know it intuitively, which we do.

I carried this problem around in my head basically the whole time. I would wake up with it first thing in the morning, I would be thinking about it all day, and I would be thinking about it when I went to sleep. Without distraction I would have the same thing going round and round in my mind. (Recalling the degree of focus and determination that eventually yielded the proof of Fermat's Last Theorem.)

Proof is what lies at the heart of maths, and is what marks it out from other sciences. Other sciences have hypotheses that are tested against experimental evidence until they fail, and are overtaken by new hypotheses. In maths, absolute proof is the goal, and once something is proved, it is proved forever, with no room for change.

There are no proofs. There are only agreements

Nowadays people rush into print before they’ve even finished the proof.

This skipping is another important point. It should be done whenever a proof seems too hard or whenever a theorem or a whole paragraph does not appeal to the reader. In most cases he will be able to go on and later he may return to the parts which he skipped.

It is a matter for considerable regret that Fermat, who cultivated the theory of numbers with so much success, did not leave us with the proofs of the theorems he discovered. In truth, Messrs Euler and Lagrange, who have not disdained this kind of research, have proved most of these theorems, and have even substituted extensive theories for the isolated propositions of Fermat. But there are several proofs which have resisted their efforts.

There is much creativity underlying math and science problem solving. Many people think that there’s only one way to do a problem, but there are often a number of different solutions, if you have the creativity to see them. For example, there are more than three hundred different known proofs of the Pythagorean theorem.

I am not qualified to say whether or not God exists. I kind of doubt He does. Nevertheless I'm always saying that the SF( The SF is the supreme Fascist, the Number-One guy up there) has this transfinite book-transfinite being a concept in mathematics that is larger than infinite-that contains the best proofs of all mathematical theorems, proofs that are elegant and perfect.

One of them, Emmanuel Giroux, is a true giant of mathematics and currently heads a laboratory of sixty people at the École normale supérieure in Lyon. Blind since the age of eleven, he is most well-known for his beautiful proof of an important theorem of contact geometry.

Instead of proving all possible theorems in an axiomatic system (which Kurt Gödel showed is not possible), professional mathematicians continue to use a formal presentation of mathematics to specify and prove many theorems that are amenable to the formalist paradigm. This has generated a vast corpus of formal theory. Controversies continue unresolved. Some mathematicians continue to insist on giving explicit constructions of mathematical entities, and do not allow proof by contradiction. This is a valid approach in its own right with much to recommend it. In the end, however, the choice that is likely to lead to the greater conquests is the one that offers the greater power and at the moment, it is David Hilbert's formalism that continues to predominate, while steadily being expanded as mathematics expands." -David Tall (2013, p. 246) thinks though Formalism (mathematics) may have Lost the Battle it Still may Win the War.

One can be enlightened about proofs as well as theorems. Without enlightenment, one is merely reduced to memorizing proofs. With enlightenment about a proof, its flow becomes clear and it can become an item of astonishing beauty. In addition, the need to memorize disappears because the proof has become part of your soul.

Some could say it is the external world which has molded our thinking-that is, the operation of the human brain-into what is now called logic. Others-philosophers and scientists alike-say that our logical thought (thinking process?) is a creation of the internal workings of the mind as they developed through evolution "independently" of the action of the outside world. Obviously, mathematics is some of both. It seems to be a language both for the description of the external world, and possibly even more so for the analysis of ourselves. In its evolution from a more primitive nervous system, the brain, as an organ with ten or more billion neurons and many more connections between them must have changed and grown as a result of many accidents. The very existence of mathematics is due to the fact that there exist statements or theorems, which are very simple to state but whose proofs demand pages of explanations. Nobody knows why this should be so. The simplicity of many of these statements has both aesthetic value and philosophical interest.

Since then, several other conjectures have been resolved with the aid of computers (notably, in 1988, the nonexistence of a projective plane of order 10). Meanwhile, mathematicians have tidied up the Haken-Appel argument so that the computer part is much shorter, and some still hope that a traditional, elegant, and illuminating proof of the four-color theorem will someday be found. It was the desire for illumination, after all, that motivated so many to work on the problem, even to devote their lives to it, during its long history. (One mathematician had his bride color maps on their honeymoon.) Even if the four-color theorem is itself mathematically otiose, a lot of useful mathematics got created in failed attempts to prove it, and it has certainly made grist for philosophers in the last few decades. As for its having wider repercussions, I’m not so sure. When I looked at the map of the United States in the back of a huge dictionary that I once won in a spelling bee for New York journalists, I noticed with mild surprise that it was colored with precisely four colors. Sadly, though, the states of Arkansas and Louisiana, which share a border, were both blue.

In 1931, Kurt Godel proved in his famous second incompleteness theorem that there could be no finitary proof of the consistency of arithmetic. He had killed Hilbert's program with a single stroke. So should you be worried that all of mathematics might collapse tomorrow afternoon? For what it's worth, I'm not. I do believe in infinite sets, and I find the proofs of consistency that use infinite sets to be convincing enough to let me sleep at night.

Science in its everyday practice is much closer to art than to philosophy. When I look at Godel's proof of his undecidability theorem, I do not see a philosophical argument. The proof is a soaring piece of architecture, as unique and as lovely as Chartres Cathedral. Godel took Hilbert's formalized axioms of mathematics as his building blocks and built out of them a lofty structure of ideas into which he could finally insert his undecidable arithmetical statement as the keystone of the arch. The proof is a great work of art. It is a construction, not a reduction. It destroyed Hilbert's dream of reducing all mathematics to a few equations, and replaced it with a greater dream of mathematics as an endlessly growing realm of ideas. Godel proved that in mathematics the whole is always greater than the sum of the parts. Every formalization of mathematics raises questions that reach beyond the limits of the formalism into unexplored territory.

Here are the basic principles of Constructivism as practiced by Kronecker and codified by J.H. Poincare and L.E.J. Brouwer and other major figures in Intuitionism: (1) Any mathematical statement or theorem that is more complicated or abstract than plain old integer-style arithmetic must be explicitly derived (i.e. 'constructed') from integer arithmetic via a finite number of purely deductive steps. (2) The only valid proofs in math are constructive ones, with the adjective here meaning that the proof provides a method for finding (i.e., 'constructing') whatever mathematical entities it's concerned with.

It is an unfortunate fact that proofs can be very misleading. Proofs exist to establish once and for all, according to very high standards, that certain mathematical statements are irrefutable facts. What is unfortunate about this is that a proof, in spite of the fact that it is perfectly correct, does not in any way have to be enlightening. Thus, mathematicians, and mathematics students, are faced with two problems: the generation of proofs, and the generation of internal enlightenment. To understand a theorem requires enlightenment. If one has enlightenment, one knows in one's soul why a particular theorem must be true.

The theorem states that in zero-sum games in which the players’ interests are strictly opposed (one’s gain is the other’s loss), one player should attempt to minimize his opponent’s maximum payoff while his opponent attempts to maximize his own minimum payoff. When they do so, the surprising conclusion is that the minimum of the maximum (minimax) payoffs equals the maximum of the minimum (maximin) payoffs. The general proof of the minimax theorem is quite complicated, but the result is useful and worth remembering. If all you want to know is the gain of one player or the loss of the other when both play their best mixes, you need only compute the best mix for one of them and determine its result.

And John Nash, my mathematical hero, revolutionized analysis and geometry with the proof of three theorems in scarcely more than five years before succumbing to paranoid schizophrenia. There is a fine line, it is often said, between genius and madness. Neither of these concepts is well defined, however. And in the case not only of Grothendieck but also of Gödel and Nash, periods of mental derangement, so far from promoting mathematical productivity, actually precluded it. Innate versus acquired, a classic debate. Fischer, Grothendieck, Erdős, and Perelman were all Jewish. Of these, Fischer and Erdős were Hungarian. No one who is familiar with the world of science can have failed to notice how many of the most gifted mathematicians and physicists of the twentieth century were Jews, or how many of the greatest geniuses were Hungarian (many

In a companion essay to "Continuity and I.N." that's usually translated as "The Nature and Meaning of Numbers," Dedekind evinces a remarkable proof for his "Theorem 66. There exist infinite systems," which runs thus: "My own realm of thoughts, i.e., the totality S of all things which can be objects of my thought, is infinite. For if s signifies an element of S, then the thought s', that s can be an object of my thought, is itself an element of S,..." and so on, meaning that the infinite series ([s] + [s is an object of thought]+['s is an object of thought' is an object of thought] + ...) exists in the Gedankenwelt, which entails that the Gedankenwelt is itself infinite. With respect to this proof, notice (a) how closely it resembles the various Zeno-like VIR back in paragraph 2a, and (b) how easily we could object that the proof establishes only that Dedekind's Gedankenwelt is 'potentially infinite' (and in precisely Aristotle's sense of the term), since nobody can ever actually sit down and think a whole infinite series of (s+s'+s")-type thoughts-i.e., the series is a total abstraction.

If something is true and you try to disprove it, you will fail. We are trained to to think of failure as bad, but it's not all bad. You can learn from failure. You try to disprove the statement one way, and you hit a wall. You try another way, and you hit another wall. Each night you try, each night you fail, each night a new wall, and if you are lucky, those walls start to come together into a structure, and that structure is the structure of the proof of the theorem. For if you have really understood what's keeping you from disproving the theorem, you very likely understand, in a way inaccessible to you before, why the theorem is true. This is what happened to Bolyai, who bucked his father's well-meaning advice and tried, like so many before him, to prove that the parallel postulate followed from Euclid's other axioms. Like all the others, he failed. But unlike the others, he was able to understand the shape of his failure. What was blocking all his attempts to prove that there was no geometry without the parallel postulate was the existence of just such a geometry! And with each failed attempt he learned more about the features of the thing he didn't think existed, getting to know it more and more intimately, until the moment when he realized it was really there.

Our mathematics is a combination of invention and discoveries. The axioms of Euclidean geometry as a concept were an invention, just as the rules of chess were an invention. The axioms were also supplemented by a variety of invented concepts, such as triangles, parallelograms, ellipses, the golden ratio, and so on. The theorems of Euclidean geometry, on the other hand, were by and large discoveries; they were the paths linking the different concepts. In some cases, the proofs generated the theorems-mathematicians examined what they could prove and from that they deduced the theorems. In others, as described by Archimedes in The Method, they first found the answer to a particular question they were interested in, and then they worked out the proof. Typically, the concepts were inventions. Prime numbers as a concept were an invention, but all the theorems about prime numbers were discoveries. The mathematicians of ancient Babylon, Egypt, and China never invented the concept of prime numbers, in spite of their advanced mathematics. Could we say instead that they just did not "discover" prime numbers? Not any more than we could say that the United Kingdom did not "discover" a single, codified, documentary constitution. Just as a country can survive without a constitution, elaborate mathematics could develop without the concept of prime numbers. And it did! Do we know why the Greeks invented such concepts as the axioms and prime numbers? We cannot be sure, but we could guess that this was part of their relentless efforts to investigate the most fundamental constituents of the universe. Prime numbers were the basic building blocks of matter. Similarly, the axioms were the fountain from which all geometrical truths were supposed to flow. The dodecahedron represented the entire cosmos and the golden ratio was the concept that brought that symbol into existence.

In any event, Socrates’ proof of prenatal immortality is that one of Meno’s uneducated slave boys actually comes up with the Pythagorean theorem without ever having studied geometry! Therefore, he must be remembering it. You recall that theorem: in a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides. Huh? We can barely remember that from tenth grade, let alone from before we were born.

In the theory of well-ordered series and compact series, we have followed Cantor closely, except in dealing with Zermelo's theorem (*257—8), and in cases where Cantor's work tacitly assumes the multiplicative axiom. Thus what novelty there is, is in the main negative. In particular, the multiplicative axiom is required in all known proofs of the fundamental proposition that the limit of a progression of ordinals of the second class {i.e. applicable to series whose fields have ^{o terms) is an ordinal of the second class (cf *265). In consequence of this fact, a very large part of the recognized theory of transfinite ordinals must be considered doubtful.

There is a somewhat surprising source of cyclic groups: if p is prime, the group ((Z/pZ) ! , ·) is cyclic. We will prove a more general statement when we have accumulated more machinery (Theorem IV.6.10), but the adventurous reader can already enjoy a proof by working out Exercise 4.11. This is a relatively deep fact; note that, for example, (Z/12Z) ! is not cyclic (cf. Exercise 2.19, and Exercise 4.10). The fact that (Z/pZ) ! is cyclic for p prime means that there must be integers a such that every non-multiple of p is congruent to a power of a; the usual proofs of this fact are not constructive, that is, they do not explicitly produce an integer with this property. There is a very pretty connection between the order of an element of the cyclic group (Z/pZ) ! and the so-called ‘cyclotomic polynomials’; but that will have to wait for a little field theory (cf. Exercise VII.5.15

What about origination in mathematics? This is also a linking, but this time of what needs to be demonstrated-usually a theorem-to certain conceptual forms or principles that will together construct the demonstration. Think of a theorem as a carefully constructed logical argument. It is valid if it can be constructed under accepted logical rules from other valid components of mathematics-other theorems, definitions, and lemmas that form the available parts and assemblies in mathematics. Typically the mathematcian "sees" or struggles to see one or two overarching principles: conceptual ideas that if provable provide the overall route to a solution. To be proved, these must be constructed from other accepted subprinciples or theorems. Each part moves the argument part of the way. Andrew Wiles' proof of Fermat's theorem uses as its base principle a conjecture by the Japanese mathematicians Taniyama and Shimura that connects two main structures he needs, modular forms and elliptic equations. To prove this conjecture and link the components of the argument, Wiles uses many subprinciples. "You turn to a page and there's a brief appearance of some fundamental theorem by Deligne," says mathematician Kenneth Ribet, "and then you turn to another page and in some incidental way there's a theorem by Hellegouarch-all of these things are just called into play and used for a moment before going on to the next idea." The whole is a concatenation of principles-conceptual ideas-architected together to achieve the purpose. And each component principle, or theorem, derives from some earlier concatenation. Each, as with technology, provides some generic functionality-some key piece of the argument-used in the overall structure. That origination in science or in mathematics is not fundamentally different from that in technology should not be surprising. The correspondences exist not because science and mathematics are the same as technology. They exist because all three are purposed systems-means to purposes, broadly interpreted-and therefore must follow the same logic. All three are constructed from forms or principles: in the case of technology, conceptual methods; in the case of science, explanatory structures; in the case of mathematics, truth structures consistent with basic axioms. Technology, scientific explanation, and mathematics therefore come into being via similar types of heuristic process-fundamentally a linking between a problem and the forms that will satisfy it.

The incompleteness theorem is a mathematical theorem precisely because the relevant notions of truth and provability are mathematically definable. Nonmathematical “Gödel sentences” and Liar sentences give rise to prolonged (or endless) discussions of just what is meant by a proof, by a true statement, by sound reasoning, by showing something to be true, by convincing oneself of something, by believing something, by a meaningful statement, and so on.

You’ll often see posts about people beating the CAP theorem. They haven’t. What they have done is create a system where some capabilities are CP, and some are AP. The mathematical proof behind the CAP theorem holds. Despite many attempts at school, I’ve learned that you don’t beat math.

You’ll often see posts about people beating the CAP theorem. They haven’t. What they have done is create a system where some capabilities are CP, and some are AP. The mathematical proof behind the CAP theorem holds. Despite many attempts at school, I’ve learned that you don’t beat math

It follows that neither the theorems of mathematics, nor the process of mathematical proof, nor the experience of mathematical intuition, confers any certainty. Nothing does.

I am living proof of the Infinite Monkey Theorem.

Turing was able to show that there are certain classes of problem that do not have any algorithmic solution (in particular the 'halting problem' that I shall describe shortly). However, Hilbert's actual tenth problem had to wait until 1970 before the Russian mathematician Yuri Matiyasevich-providing proofs that completed certain arguments that had been earlier put forward by the Americans Julia Robinson, Martin Davis, and Hilary Putnam-showed that there can be no computer program (algorithm) which decides yes/no systematically to the question of whether a system of Diophantine equations has a solution. It may be remarked that whenever the answer happens to be 'yes', then that fact can, in principle, be ascertained by the particular computer program that just slavishly tries all sets of integers one after the other. It is the answer 'no', on the other hand, that eludes any systematic treatment. Various sets of rules for correctly giving the answer 'no' can be provided-like the argument using even and odd numbers that rules out solutions to the second system given above-but Matisyasevich's theorem showed that these can never be exhaustive.

One thing that we conclude from all this is that the 'learning robot' procedure for doing mathematics is not the procedure that actually underlies human understanding of mathematics. In any case, such bottom-up-dominated procedure would appear to be hopelessly bad for any practical proposal for the construction of a mathematics-performing robot, even one having no pretensions whatever for simulating the actual understandings possessed by a human mathematician. As stated earlier, bottom-up learning procedures by themselves are not effective for the unassailable establishing of mathematical truths. If one is to envisage some computational system for producing unassailable mathematical results, it would be far more efficient to have the system constructed according to top-down principles (at least as regards the 'unassailable' aspects of its assertions; for exploratory purposes, bottom-up procedures might well be appropriate). The soundness and effectiveness of these top-down procedures would have to be part of the initial human input, where human understanding an insight provide the necesssary additional ingredients that pure computation is unable to achieve. In fact, computers are not infrequently employed in mathematical arguments, nowadays, in this kind of way. The most famous example was the computer-assisted proof, by Kenneth Appel and Wolfgang Haken, of the four-colour theorem, as referred to above. The role of the computer, in this case, was to carry out a clearly specified computation that ran through a very large but finite number of alternative possibilities, the elimination of which had been shown (by the human mathematicians) to lead to a general proof of the needed result. There are other examples of such computer-assisted proofs and nowadays complicated algebra, in addition to numerical computation, is frequently carried out by computer. Again it is human understanding that has supplied the rules and it is a strictly top-down action that governs the computer's activity.

There is one are of work that should be mentioned here, referred to as 'automatic theorem proving'. One set of procedures that would come under this heading consists of fixing some formal system H, and trying to derive theorems within this system. We recall, from 2.9, that it would be an entirely computational matter to provide proofs of all the theorems of H one after the other. This kind of thing can be automated, but if done without further thought or insight, such an operation would be likely to be immensely inefficient. However, with the employment of such insight in the setting up of the computational procedures, some quite impressive results have been obtained. In one of these schemes (Chou 1988), the rules of Euclidean geometry have been translated into a very effective system for proving (and sometimes discovering) geometrical theorems. As an example of one of these, a geometrical proposition known as V. Thebault's conjecture, which had been proposed in 1938 (and only rather recently proved, by K.B. Taylor in 1983), was presented to the system and solved in 44 hours' computing time. More closely analogous to the procedures discussed in the previous sections are attempts by various people over the past 10 years or so to provide 'artificial intelligence' procedures for mathematical 'understanding'. I hope it is clear from the arguments that I have given, that whatever these systems do achieve, what they do not do is obtain any actual mathematical understanding! Somewhat related to this are attempts to find automatic theorem-generating systems, where the system is set up to find theorems that are regarded as 'interesting'-according to certain criteria that the computational system is provided with. I do think that it would be generally accepted that nothing of very great actual mathematical interest has yet come out of these attempts. Of course, it would be argued that these are early days yet, and perhaps one may expect something much more exciting to come out of them in the future. However, it should be clear to anyone who has read this far, that I myself regard the entire enterprise as unlikely to lead to much that is genuinely positive, except to emphasize what such systems do not achieve.

I walked around those places, pious child that i was, thinking that my goodness was proof negative. "Look at me!" I wanted to shout. I wanted to be a living theorem, a Logos. Science and math had already taught me that if there was many exceptions to a rule, then the rule was not a rule. Look at me. This was all so wrongheaded, so backward, but I didn't know how to think any differently. The rule was never a rule, but I had mistaken it for one. I took me years of questioning and seeking to see more than my little piece, and even now I don't always see it.

The Banach-Tarski Theorem is an astonishing result. We have decomposed a ball into finitely many pieces, moved around the pieces without changing their size or shape, and then reassembled them into two balls of the same size as the original. I think the theorem teaches us something important about the notion of volume. As noted earlier, it is an immediate consequence of the theorem that some of the Banach-Tarski pieces must lack definite volumes and, therefore, that not every subset of the unit ball can have a well-defined volume. A little more precisely, the theorem teaches us that there is no way of assigning volumes to the Banach-Tarski pieces while preserving three-dimensional versions of the principles we called Uniformity and (finite) Additivity in chapter 7. (Proof: Suppose that each of the (finitely many) Banach-Tarski pieces has a definite finite volume. Since the pieces are disjoint, and since their union is the original ball, Additivity entails that the sum of the volumes of the pieces must equal the volume of the original ball. But Uniformity ensures that the volume of each piece is unchanged as we move it around. Since the reassembled pieces are disjoint, and since their union is two balls, Additivity entails that the sum of their volumes must be twice the volume of the original ball. But since the volume of the original ball is finite and greater than zero, it is impossible for the sum of the pieces to equal both the volume of the original ball and twice the volume of the original ball.) If I were to assign the Banach-Tarski Theorem a paradoxicality grade of the kind we used in chapter 3, I would assign it an 8. The theorem teaches us that although the notion of volume is well-behaved when we focus on ordinary objects, there are limits to how far it can be extended when we consider certain extraordinary objects - objects that can only be shown to exist by assuming the Axiom of Choice.

We do not want many ‘variations’ in the proof of a mathematical theorem: ‘enumeration of cases’, indeed, is one of the duller forms of mathematical argument. A mathematical proof should resemble a simple and clear-cut constellation, not a scattered cluster in the Milky Way.

A theory, a theorem and a hypothesis walk into a bar, but leave as soon as the bartender asks them for proofs.

Of course, no work in this genre would be complete without an allusion to Gödel’s theorem: This drifting of figures and geometric figuring, this irruption of dimensions and transcendental mathematics, leads us to the promised surrealist peaks of scientific theory, peaks that culminate in Gödel’s theorem: the existential proof, a method that mathematically proves the existence of an object without producing that object. (Virilio 1991, p. 66) In reality, existential proofs are much older than Gödel’s work; and the proof of his theorem is, by contrast, completely constructive: it exhibits a proposition that is neither provable nor falsifiable

That might sound strange, coming from me. I know how I'm seen. I'll be remembered for my theorem, for my mind. People don't see my greatest gift was really...my heart.

language that felt linked to my experience. In the sixties, the physicist John Stewart Bell theorized that particles that were once connected will, when separated, behave as if still connected, regardless of the distance between them. Some years later a French physicist, Alain Aspect, conducted experiments offering physical proof of Bell’s theorem.

The suggestion that eternal recurrence might be proved as a theorem of physics, rather than as a religious or philosophical doctrine, seems to have occurred at about the same time to the German philosopher Friedrich Nietzsche and the French mathematician Henri Poincaré. Nietzsche encountered the idea of recurrence on his studies of classical philology, and again in a book by Heine. It was not until 1881 that he began to take it seriously, however, and then he devoted several years to studying physics in order to find a scientific-sounding formulation of it. Poincaré on the other hand, was led to the subject by his attempts to complete Poisson's proof of the stability of the solar system, though he was also concerned with the difficulty of explaining irreversibility by mechanical models such as Helmholtz's monocyclic systems. Poincaré's theorem belongs to the history of theoretical physics, Nietzsche's speculations to the history of philosophical culture, and they are not usually discussed in the same context. Yet I find it necessary to consider them together since it was just at the end of the 19th century that developments in science were strongly coupled to the philosophical-cultural background. Both Nietzsche and Poincaré were trying, though in very different ways, to attack the "materialist" or "mechanist" view of the universe.

Of a totally different orientation [from the "Old Formalist School" of Dedekind, Cantor, Peano, Zermelo, and Couturat, etc.] was the Pre-Intuitionist School, mainly led by Poincaré, Borel and Lebesgue. These thinkers seem to have maintained a modified observational standpoint for the introduction of natural numbers, for the principle of complete induction [...] For these, even for such theorems as were deduced by means of classical logic, they postulated an existence and exactness independent of language and logic and regarded its non-contradictority as certain, even without logical proof. For the continuum, however, they seem not to have sought an origin strictly extraneous to language and logic.

In fact, so detailed was his interest in mathematics, and so acute his understanding, that he had recently written an original proof of the Pythagorean Theorem during a free moment at the capital. The New England Journal of Education had published the proof just the month before, transparently astonished that a member of congress had written it. Despite Garfield's deep admiration for mathematics and the arts, however, he believed that it was science, above all other disciplines that had achieved the greatest good.

The basic point of all the scientific ideas we threw at you is that there is a lot of disagreement about how the flow of time works and how or whether one thing causes another. If you take home one idea out of all of these, make it that the everyday feeling that the future has no effect on the present is not necessarily true. As a result of the current uncertainty about time and causality in philosophical and scientific circles, it is not at all unreasonable to talk in a serious way about the possibility of genuine precognition. We also hope that our brief mention of spirituality has opened your mind to the idea that there may be a spiritual perspective as well. Both Theresa and Julia treasure the spiritual aspects of precognition, because premonitions can act as reminders that there may be an eternal part of us that exists outside of time and space. There may well be a scientific explanation for this eternal part, and if one is found, science and spirituality will become happy partners. Much of Part 2 will be devoted to the spiritual and wellbeing components of becoming a Positive Precog, and we will continue to marry those elements with scientific research as we go. 1 Here, physics buffs might chime in with some concerns about the Second Law of Thermodynamics. Okay, physics rock stars! As you know, the Second Law states that in a closed system, disorder is very unlikely to decrease – and as such, you may believe this means that there is an “arrow of time” that is set by the Second Law, and this arrow goes in only the forward direction. As a result, you might also think that any talk of a future event influencing the past is bogus. We would ask you to consider four ideas. 2 Here we are not specifically talking about closed timelike curves, but causal loops in general. 3 For those concerned that the idea of messages from the future suggests such a message would be travelling faster than the speed of light, a few thoughts: 1) “message” here is used colloquially to mean “information” – essentially a correlation between present and future events that can’t be explained by deduction or induction but is not necessarily a signal; 2) recently it has been suggested that superluminal signalling is not actually prohibited by special relativity (Weinstein, S, “Superluminal signaling and relativity”, Synthese, 148(2), 2006: 381–99); and 3) the no-signalling theorem(s) may actually be logically circular (Kennedy, J B, “On the empirical foundations of the quantum no-signalling proofs”, Philosophy of Science, 62(4), 1995: 543–60.) 4 Note that in the movie Minority Report, the future was considered set in stone, which was part of the problem of the Pre-Crime Programme. However, at the end of the movie it becomes clear that the future envisioned did not occur, suggesting the idea that futures unfold probabilistically rather than definitely.

The fact of zero He added nonstop: Did you know that zero was not used throughout human history! Until 781 A.D, when it was first embodied and used in arithmetic equations by the Arab scholar Al-Khwarizmi, the founder of algebra. Algorithms took their name from him, and they are algorithmic arithmetic equations that you have to follow as they are and you will inevitably get the result, the inevitable result. And before that, across tens and perhaps hundreds of thousands of years, humans refused to deal with zero. While the first reference to it was in the Sumerian civilization, where inscriptions were found three thousand years ago in Iraq, in which the Sumerians indicated the existence of something before the one, they refused to deal with it, define it and give it any value or effect, they refused to consider it a number. All these civilizations, some of which we are still unable to decipher many of their codes, such as the Pharaonic civilization that refused to deal with zero! We see them as smart enough to build the pyramids with their miraculous geometry and to calculate the orbits of stars and planets with extreme accuracy, but they are very stupid for not defining zero in a way that they can deal with, and use it in arithmetic operations, how strange this really is! But in fact, they did not ignore it, but gave it its true value, and refused to build their civilizations on an unknown and unknown illusion, and on a wrong arithmetical frame of reference. Throughout their history, humans have looked at zero as the unknown, they refused to define it and include it in their calculations and equations, not because it has no effect, but because its true effect is unknown, and remaining unknown is better than giving it a false effect. Like the wrong frame of reference, if you rely on it, you will inevitably get a wrong result, and you will fall into the inevitability of error, and if you ignore it, your chance of getting it right remains. Throughout their history, humans have preferred to ignore zero, not knowing its true impact, while we simply decided to deal with it, and even rely on it. Today we build all our ideas, our civilization, our software, mathematics, physics, everything, on the basis that 1 + 0 equals one, because we need to find the effect of zero so that our equations succeed, and our lives succeed with, but what if 1 + 0 equals infinity?! Why did we ignore the zero in summation, and did not ignore it in multiplication?! 1×0 equals zero, why not one? What is the reason? He answered himself: There is no inevitable reason, we are not forced. Humans have lived throughout their ages without zero, and it did not mean anything to them. Even when we were unable to devise any result that fits our theorems for the quotient of one by zero, then we admitted and said unknown, and ignored it, but we ignored the logic that a thousand pieces of evidence may not prove me right, and one proof that proves me wrong. Not doing our math tables in the case of division, blowing them up completely, and with that, we decided to go ahead and built everything on that foundation. We have separated the arithmetic tables in detail at our will, to fit our calculations, and somehow separate the whole universe around us to fit these tables, despite their obvious flaws. And if we decide that the result of one multiplied by zero is one instead of zero, and we reconstruct the whole world on this basis, what will happen? He answered himself: Nothing, we will also succeed, the world, our software, our thoughts, our dealings, and everything around us will be reset according to the new arithmetic tables. After a few hundred years, humans will no longer be able to understand that one multiplied by zero equals zero, but that it must be one because everything is built on this basis.

Torricelli fully realized the advantages and disadvantages of the method of indivisibles; and he suspected that the ancients possessed some such method for discovering difficult theorems, the proofs of which they cast in another form either "to hide the secret of their method or to avoid giving occasion for contradiction to jealous detractors.

wide range of behavioral relations, and is now an umbrella term for the science of logical decision making in humans, animals, and computers. Modern game theory began with the idea regarding the existence of mixed-strategy equilibria in two-person zero-sum games and its proof by John von Neumann. Von Neumann's original proof used the Brouwer fixed-point theorem on continuous mappings into compact convex sets, which became a standard method in game theory and mathematical economics. His paper was

mathematics as a tree with strong roots (the Axioms), a solid trunk (Rigorous Proof) and ever growing branches blooming with wondrous flowers (the Theorems).

Proof of the beginning of time probably ranks as the most theologically significant theorem. This great significance arises from the theorem establishing that the universe must be caused by some Entity capable of creating the universe entirely independent of space and time. Such an entity matches the attributes of the God of the Bible but is contradicted by the gods of the eastern (and indeed all other) religions who create within space and time.

what you are doing is sort of architectural. You have to have a design in view, in which you design a chapter, or a proof of a theorem, as the case may be. Then you have to put it together out of words or out of symbols as the case may be, but if you don’t have a clear architecture in mind then the thing won’t end up being any good.

Fermat’s Last Theorem dates to 1637. The French mathematician and physicist Pierre de Fermat had scribbled it in the margins of a book, adding that he had discovered a marvelous proof but that the margins were too small to hold it.

The construction of a "problem calculus" in the sense of Heyting and Kolmogoroff yields a model of logic in which the theorem of the excluded middle does not appear among the basic formulas. The study of such a logic widens our insight into the basic elements of mathematics and, in particular, points out the special position of the so-called indirect proofs within mathematics.

B: True. I've got this made urge to get up before a class and present our results: Theorem, proof, lemma, remark. I'd make it so slick nobody would be able to guess how we did it, and everyone would be

B: True. I've got this mad urge to get up before a class and present our results: Theorem, proof, lemma, remark. I'd make it so slick nobody would be able to guess how we did it, and everyone would be so impressed.

Undoubtedly, the capstone of every mathematical theory is a convincing proof of all of its assertions. Undoubtedly, rnathematics inculpates itself when it foregoes convincing proofs. But the mystery of brilliant productivity will always be the posing of new questions, the anticipation of new theorems that make accessible valuable results and connections. Without the creation of new viewpoints, without the statement of new aims, mathematics would soou exhaust itself in the rigor of its logical proofs and begin to stagnate as its substance vanishes. Thus, in a sense, mathematics has been most advanced by those who distinguished thernselves by intuition ratber than by rigorous proofs.