The Outer Limits of Reason Quotes

“Douglas R. Hofstadter, an American researcher, speculates that the human mind has consciousness because it has the capability of self-reference. Since we can think about ourselves and think about ourselves thinking about ourselves, etc., we are capable of feeling that we are an "I". Contrast that with what we have learned in this chapter. This chapter tries to show that the computer's ability to perform self-reference is the cause of its limitations. Can we say that self-reference in computers brings limitations while in humans it causes consciousness? Perhaps.”
― Noson S. Yanofsky, The Outer Limits of Reason: What Science, Mathematics, and Logic Cannot Tell Us

“the universe uses scientists to study itself.”
― Noson S. Yanofsky, The Outer Limits of Reason: What Science, Mathematics, and Logic Cannot Tell Us

“Within environments capable of sustaining humans, there are constant tsunamis, volcanoes, earthquakes, hurricanes, mudslides, poisonous mushrooms, and lawyers, all of which make human life painfully fragile.”
― Noson S. Yanofsky, The Outer Limits of Reason: What Science, Mathematics, and Logic Cannot Tell Us

“The Banach-Tarski paradox The pieces that the original ball is chopped into are not your typical sane pieces. Rather, they will look like something that was done by Zeno while under the influence of psychedelic drugs. Each piece will be connected but very bizarre looking. Nevertheless, this fact is a provable consequence of the seemingly harmless axiom of choice. Another version of the paradox says that a sphere as small as a pea can be cut up into a finite set of different parts and then be put together to form a sphere the size of the sun. Many people say that since this paradox is the consequence of the axiom of choice, this axiom leads to an obvious false statement and should be excluded from what is reasonable.”
― Noson S. Yanofsky, The Outer Limits of Reason: What Science, Mathematics, and Logic Cannot Tell Us

“One of the more interesting paradoxes about knowledge is called the surprise-test paradox. A teacher announces that there will be a surprise test in the forthcoming week. The last day of class is Friday of that week. What day can the surprise test happen on? If the test is going to be on Friday, then after school on Thursday night the students will already know that the test is on Friday and it will not be a surprise test. So the test cannot happen on Friday. Since this was purely logical reasoning, everyone knows this. Can the test be on Thursday? After class on Wednesday night the students can deduce that since the test has not happened already and it cannot be on Friday, it must be on Thursday. But again, since they know that it must be on Thursday, it will no longer be a surprise test. So the test cannot occur on Thursday or Friday. We can continue reasoning in the same way and conclude that the test cannot happen on Wednesday, Tuesday, or Monday. When exactly will this surprise test occur? Logic has shown us that a teacher cannot give a surprise test within a given time interval. This is a paradox because it goes against the obvious fact that teachers have been torturing students with surprise tests for millennia. It”
― Noson S. Yanofsky, The Outer Limits of Reason: What Science, Mathematics, and Logic Cannot Tell Us

“There is an urge to simply wave away all this talk of relativity and insist on absolute space and time. One wants to merely declare that the measurements done from a stationary position on Earth are the absolute measurements and every other measurement is relative. This would be an error. Although it seems like the Earth is not moving, it is , in fact, constantly moving in a wild pattern. Remember that the Earth is spinning on its axis at about 1,000 miles per hour. It is rotating around the sun at about 67,000 miles per hour. Furthermore, our solar system is moving around our galaxy at about a half a million miles per hour. Poke your finger into the air. Wait a second. Now poke your finger "in the same place." Realize that the two places where you poked your finger are hundreds, if not thousands, of miles apart. A stationary observer on Earth is far from stationary. There are no absolute observers, no absolute measurements, and no absolute space and time. All is relative.”
― Noson S. Yanofsky, The Outer Limits of Reason: What Science, Mathematics, and Logic Cannot Tell Us

“Physicists have formulated equations that indicate exactly how much length contraction objects will experience and how slow time will progress relative to the stationary observer. These equations take into account the velocity of the observer. The faster the movement, the more space contraction and time dilation will occur. What is the limit of this process? What if people could go very, very fast? If they were actually able to go the speed of light, they would shrink to nothingness and time would totally stop for them. That is, they could not exist. This is yet another consequence of special relativity: there is a type of cosmic speed limit. Nothing can move as fast as, or faster than, the speed of light. This is, quite literally, a limitation described by science.”
― Noson S. Yanofsky, The Outer Limits of Reason: What Science, Mathematics, and Logic Cannot Tell Us

“For now, the correct interpretation of quantum mechanics is beyond science.”
― Noson S. Yanofsky, The Outer Limits of Reason: What Science, Mathematics, and Logic Cannot Tell Us

“I conclude with a little meditation on the number of physical phenomena that can and cannot be explained/predicated by science. In a sense, language, be it spoken or written, be it natural language or exact formulas, is countably infinite. There is no longest word or longest novel, because there is no limit to the longest formula, and so on. This makes language infinite. However, it can be alphabetized or counted, which makes language countably infinite. In contrast to language, which can be used to describe or predict phenomena, let us examine what is really "out there." It is plausible to say that there is an uncountably infinite number of phenomena that can occur. This is stated without proof because I cannot quantify all phenomena. To quantify them, I would have to describe them and I cannot do that without language. So there might be an uncountably infinite number of phenomena and only a small, countably infinite subset describable by science. This is the ultimate, nonscientific (science must stay within the bounds of language) limitation on science's ability. At this point we must take Wittgenstein's dictum to heart: "What we cannot speak about we must pass over in silence.”
― Noson S. Yanofsky, The Outer Limits of Reason: What Science, Mathematics, and Logic Cannot Tell Us

“As you may have noticed, all the problems that we have so far shown to be undecidable have to do with determining different properties about programs. In other words, we have shown that there are no programs to determine certain properties about programs. This follows our theme that there are limitations when there is self-reference. In 1951, Henry Rice proved the granddaddy of all such theorems. In what has come to be known as Rice's theorem, it was shown that there is no interesting property about programs that can be determined by a program. This rather sophisticated result is proved by showing that for any interesting property P,

Halting problem is less than or equal to Property P Problem.

Since the Halting Problem is undecidable, the Property P Problem is also undecidable.”
― Noson S. Yanofsky, The Outer Limits of Reason: What Science, Mathematics, and Logic Cannot Tell Us

“An interesting class of problems consists of those that demand a polynomial amount of space. This class is denoted as PSPACE. It is known that NP, the set of problems that can be solved in exponential or factorial time, can be solved in polynomial space. In other words, NP is a subset of PSPACE.”
― Noson S. Yanofsky, The Outer Limits of Reason: What Science, Mathematics, and Logic Cannot Tell Us

“How should you go about claiming your award money? There are two possible directions to take. You can try to prove that P = NP or you can aim to show that P is not equal to NP. To show that P = NP, all you have to do is take one of your favorite NP-Complete problems and find a polynomial algorithm that solves it. As we have seen, if you do find such an algorithm, then all NP problems will be solvable in a polynomial amount of operations. It might seem strange to think that a problem that demands an exponential or factorial amount of operations can be done in a polynomial amount of operations. It might seem strange to think that a problem that demands an exponential or factorial amount of operations can be done in a polynomial amount of operations. However, we saw something similar with the Euler Cycle Problem. Rather than look through all n! possible cycles to see if any are Euler cycles, we used the trick of checking if the number of edges touching each vertex is even or not. Does a similar trick for the Hamiltonian Cycle Problem exist? For many years, the smartest people around have been looking for such a trick or algorithm and have not been successful. However, you might possess some deeper insight that they lack. Get to it!

On the other hand, you can try to show that P is not equal to NP. One way to do this is to take an NP problem and show that no polynomial algorithm exists for it. It so happens that it is very hard to prove such a claim: there are a lot of algorithms out there. This has turned out to be one of the hardest problems in mathematics. As a final hint, it should be noted that most researchers believe that P is not equal to NP.”
― Noson S. Yanofsky, The Outer Limits of Reason: What Science, Mathematics, and Logic Cannot Tell Us

“In summary, all foreseeable future improvements to computer technology are essentially impotent in the face of these NP problems. The only way that these problems will be easily solved is to find nice polynomial algorithms for them. We will show in the next section why most researchers believe that there are no better algorithms for these problems. It looks as though they will remain problems that cannot be solved in a reasonable amount of time. These problems are not hard because we lack the technology to solve them. Rather they are hard because of the nature of the problems themselves. They are inherently hard and will probably remain on the outer limits of what we can solve.”
― Noson S. Yanofsky, The Outer Limits of Reason: What Science, Mathematics, and Logic Cannot Tell Us

“One issue that bothers both camps is the incredible usefulness of mathematics and set theory in the physical sciences. Why does the physical world somehow conform to the ideas of mathematicians and set theorists? The Platonists say that there is some type of (mysterious) connection between the Platonic realm of ideas and our physical world. They also posit some type of (mysterious) connection between the Platonic realm and our minds that permit us to discover these Platonic ideals. In contrast, nominalists say that the reason mathematics works so well is that mathematics was a language formed by humans with the intuition they received from the physical world. To them, it is not shocking that a system developed while observing the physical world should be suited to the physical world.”
― Noson S. Yanofsky, The Outer Limits of Reason: What Science, Mathematics, and Logic Cannot Tell Us

“The strongest arguments for nominalism are questions like the following: Who set up these Platonic ideals? Why are they there? For the past several hundred years, scientists have made steady progress by eliminating metaphysical presuppositions. Why keep any such metaphysics in mathematics and in set theory? A nominalist would counter a Platonist's proof by saying that the mathematicans are not all isolated from each other. Before they entered their lonely writer's garret, they were all aware of the rules for being a good mathematician. They knew that if they were to write anything that would cause a contradiction, they would lose their status as a mathematician. They were not isolated because they knew the language beforehand.”
― Noson S. Yanofsky, The Outer Limits of Reason: What Science, Mathematics, and Logic Cannot Tell Us

“The strongest argument for Platonism is the amazing consistency of mathematics. For thousands of years, mathematicians have been working in isolation from each other and have come to similar, noncontradictory ideas. It seems that the only way this is possible is that they are all trying to describe something that is external to their mind.”
― Noson S. Yanofsky, The Outer Limits of Reason: What Science, Mathematics, and Logic Cannot Tell Us

“What are we to do with all these questions? Is Zermelo-Fraenkel set theory consistent or not? Is the continuum hypothesis true or false? Is the axiom of choice acceptable or unacceptable? These questions are independent of Zermelo-Fraenkel set theory, which is a basis of most of mathematics. So we cannot use mathematics to answer these questions. The answers to all of these simply stated questions are beyond contemporary mathematics, beyond rational thought, and perhaps beyond us.”
― Noson S. Yanofsky, The Outer Limits of Reason: What Science, Mathematics, and Logic Cannot Tell Us

“Several infinite sets with cardinality Aleph 0 have been described. We have also seen several infinite sets with cardinality 2^ Aleph 0. The obvious question is whether there is anything strictly larger than 2 ^ Aleph 0. The answer is yes. The powerset of a set is strictly larger than the set. From this we can see that the powerset of (0,1), denoted powerset((0,1)), will not be in correspondence with (0,1). That is, the set of subsets of the unit interval (0,1) will be larger than (0,1). This set will have cardinality 2 ^ 2 Aleph 0. It is difficult to wrap one's mind around such a set. Try to write down some of the elements.

Of course, there is no reason to stop there. We can go on using the powerset function to describe sets of even higher cardinality. None of the sets in different levels of infinity can be put into correspondence with each other.”
― Noson S. Yanofsky, The Outer Limits of Reason: What Science, Mathematics, and Logic Cannot Tell Us

“It was shown that both (0,1) and the powerset of (N) are larger than N. In fact, a correspondence exists (which I will not describe) between these two sets showing that they have the same cardinality. Since the cardinality of the powerset of a set of size n is 2^n, and the cardinality of N is Aleph 0, cardinality of the powerset of (N) is 2^Aleph 0. Because this is also the cardinality of the continuous interval (0,1), it is also called the "cardinality of the continuum.”
― Noson S. Yanofsky, The Outer Limits of Reason: What Science, Mathematics, and Logic Cannot Tell Us

“In conclusion, many sets that seem infinitely larger than the natural numbers are, in fact, equinumerous with the natural numbers. Is there any infinite set that is actually larger than the natural numbers?”
― Noson S. Yanofsky, The Outer Limits of Reason: What Science, Mathematics, and Logic Cannot Tell Us

“One needs exact terms to do science, logic, and mathematics. When we leave the domain of exact definitions-that is, when we talk about baldness, tallness, and redness-we are necessarily leaving the boundaries where logic and math can help us. Vagueness is beyond the boundaries of reason. While we all freely live and communicate with such terms on a daily basis we must, nevertheless, be careful about crossing the outer limits of reason.”
― Noson S. Yanofsky, The Outer Limits of Reason: What Science, Mathematics, and Logic Cannot Tell Us

“Again, there is a problem abandoning the notion of continuous time for discrete time. Modern physics and engineering are based on the fact that time is continuous. All the equations have a continuous-time variable usually denoted by t. And yet, as Zeno has shown us, the notion of continuous time is illogical.”
― Noson S. Yanofsky, The Outer Limits of Reason: What Science, Mathematics, and Logic Cannot Tell Us

“If we assume the world is discrete, the mathematics needed to build rockets and bridges is far more complicated than calculus. Perhaps calculus is simply an easy approximation of the true mathematics that has to be done to concretely model the discrete world in which we live.”
― Noson S. Yanofsky, The Outer Limits of Reason: What Science, Mathematics, and Logic Cannot Tell Us

“What Zeno is forcing us to do is to ask the question of whether space (which is not made of atoms) can be infinitely divvied up. If it can be, the slacker will not reach his goal. If it cannot be, there must be discrete "space atoms," and continuous real-number mathematics is not a proper model for space.

We cannot, however be so flippant about asserting that space is discrete and not continuous. The world certainly does not look discrete. Movement has the feel of being continuous. Much of mathematical physics is based on calculus, which assumes that the real world is infinitely divisible. Outside of some quantum theory and Zeno, the continuous real number make a good model for the physical world. We build rockets and bridges using mathematics that assumes that the world is continuous. Let us not be so quick to abandon it.”
― Noson S. Yanofsky, The Outer Limits of Reason: What Science, Mathematics, and Logic Cannot Tell Us

“Another possible solution to paradoxical sentences was mentioned in chapter 1, namely, human language is a product of the human mind and as such, subject to contradictions. Human language is not a perfect system that is free of discrepancies (in contrast to perfect systems like mathematics, science, logic, and the physical universe). Rather, we should simply accept the fact that human language is faulty and has contradictions. This seems reasonable to me.”
― Noson S. Yanofsky, The Outer Limits of Reason: What Science, Mathematics, and Logic Cannot Tell Us

“Common human language has always dealt with some type of self-reference without problem:

Someone says, "Oh! I am groggy today and I do not know what I am talking about." Is he aware of saying this sentence.

Carly Simon sings a song with the lyrics "You're so vain, you probably think this song is about you." But this song is about him!

"Every rule has an exception except one rule: this one."

"Never say 'never'!"

"The only rule is that there is no rule."

In all of these cases-and many more-human language is violating the restriction of only dealing with sentences that are "below" it. In each case, a sentence discusses itself. And yet, somehow, all these examples are a legitimate part of human language.”
― Noson S. Yanofsky, The Outer Limits of Reason: What Science, Mathematics, and Logic Cannot Tell Us

“The miracle of the appropriateness of the language of mathematics for the formulation of the laws of physics is a wonderful gift which we neither understand nor deserve. We should be grateful for it and hope that it will remain valid in future research and”
― Noson S. Yanofsky, The Outer Limits of Reason: What Science, Mathematics, and Logic Cannot Tell Us

“Why is mathematics so essential to an understanding of the physical world? Why does math work so well? Why does the physical world obey mathematics? These”
― Noson S. Yanofsky, The Outer Limits of Reason: What Science, Mathematics, and Logic Cannot Tell Us

“Perhaps we are being a bit presumptuous in calling our species “intelligent.” After all, this species has waged numerous inane wars where millions of their own were slaughtered. As a whole, this species spends trillions of hours a year watching insipid television shows. And “intelligent” is not the right name for a species that invented spam e-mails and encourages narcissistic pastimes like Facebook. Nevertheless, over the millennia, this species produced many shining lights that make us worthy of the lofty title: Blaise Pascal, Isaac Newton, David Hume, Marie Curie, Albert Einstein, Arthur Stanley Eddington, Emmy Noether, Andrew Lloyd Webber, Meryl Streep, and, of course, tiramisu.”
― Noson S. Yanofsky, The Outer Limits of Reason: What Science, Mathematics, and Logic Cannot Tell Us

“It is important to realize that ideas about infinity are not abstract scholastic thoughts that plague absentminded professors in the ivy-covered towers of academia. Rather, all of calculus is based on the modern notions of infinity mentioned in this chapter. Calculus, in turn, is the basis of all of the modern mathematics, physics, and engineering that make our advanced technological civilization possible. The reason the counterintuitive ideas of infinity are central to modern science is that they work. We cannot simply ignore them.”
― Noson S. Yanofsky, The Outer Limits of Reason: What Science, Mathematics, and Logic Cannot Tell Us