The Outer Limits of Reason: What Science, Mathematics, and Logic Cannot Tell Us (Yanofsky, Noson S.)
Highlights
However, we cannot simply solve all problems by waving our hand and declaring that the word heterological does not exist or has no meaning. The problem is too deeply rooted in the very nature of language.
• “The villager who shaves everyone who does not shave themselves.” • “The word that describes all words that do not describe themselves.” • “The reference book that lists all books that do not list themselves.” • “The set that contains all sets that do not contain themselves.” As you can see, all these descriptions have the exact same structure (as do figures 2.1 through 2.3). Every time there is self-reference, there are possibilities for contradictions.
Such questions go on indefinitely. I will restrain myself and discuss just one more scenario. Imagine that every time a plank is changed, rather than consigning the old planks to the scrap heap, we store them in a warehouse. After some time, all the old planks are assembled into a ship.
Barings Bank was in existence from 1762 through 1995. In that time, the owners, workers, and customers all changed. The Brooklyn Dodgers have been around since 1883. Their players, managers, owners, and fans have definitely changed. What remains the same about a baseball team?
We are not only talking about change. Rather, we are discussing what it means for an object to be that object.
Our bodies are in constant flux. Old cells die and new cells are constantly being born. In fact, most of the cells in our body are replaced every seven years.
A person is more than a physical object because there is thought.
To such philosophers, a person is a continuous stream of consciousness—they are memories, intentions, thoughts, and desires.
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.
With this knowledge in mind, engineers and physicists build bridges and rockets. It would be foolhardy to cross a modern suspension bridge if you knew that the engineer did not believe in Cantor’s work.
“Nowadays, it is known to be possible, logically speaking, that current mathematics, almost in its entirety, can be derived from a single source: the theory of sets.”
From this important position, the axioms of Zermelo-Fraenkel set theory can be seen as the axioms of all of mathematics and hence the axioms of exact reasoning itself.
The bad news is that one of the consequences of Gödel’s famous incompleteness theorems (which we will meet, in detail, in sections 9.4 and 9.5) is that the consistency of Zermelo-Fraenkel set theory is not provable within standard mathematics.
And so we cannot be absolutely certain that Zermelo- Fraenkel set theory and all of the modern mathematics that it entails is consistent.8
They can be partitioned into fifty nonoverlapping sets that correspond to the fact that people live in fifty different states.9 We may ask for a single set of citizens that has exactly one member or representative for each subset. The simplest way of forming this set is to choose the governor of each state as the representative of that state.
However, what if we are given a partition on an infinite set? Can we still choose one element of each subset? Things seem a little more complicated in the infinite case. Imagine being presented with an infinite set of pairs of shoes. (For some, this would result in an infinite amount of joy.) We may ask for one shoe from every one of the infinite pairs. This can simply be done by always choosing the left shoe in every pair. One may also choose the right shoe in every pair. However, what if we are presented with an infinite set of pairs of tube socks where each sock is identical to its mate? Can we still choose one from each pair? Which one?
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. They want to banish the axiom of choice from mathematics.
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 even beyond us.
These battles have raged for millennia with no apparent victor. To me, any argument that Platonists give can be answered better by the nominalists. However, I am aware that we, mere mortals, are not going to come to any firm conclusions.
We are right back to the liar paradox. We have come to a point where a program halts if and only if it does not halt. Human language and human minds might have contradictions but computers do not. Something must be wrong. Only one assumption was made: that it is possible to write a program that decides the Halting Problem. That assumption caused us to reach a contradiction, hence it must have been wrong. It is impossible to solve the Halting Problem with a computer.
Penrose goes on to speculate that perhaps the brain uses the mysterious concept of quantum gravity to explain the seeming ability of humans to perform tasks that machines cannot.
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. Do human beings really have self-reference? Do we really know what is going on inside our minds?
Twas brillig, and the slithy toves Did gyre and gimble in the wabe
Chaotic systems force us to make a distinction between determinism and predictability. Determinism is a fact about the existence of laws of nature, whereas predictability is about the ability of human beings to know the future.
All that is known is that when we examine the results of a quantum experiment, or to use the right lingo, when the system is measured, we no longer see a superposition. We say the system collapses from a superposition of many positions to one particular position. The measurement problem asks why this collapse occurs and is one of the major discussion points in the philosophy of quantum mechanics.
After all, how can the photon “know” the setup of the entire experiment when it leaves the source? There might be some real distance between the source of the photons and the slits. There is no real answer to that mystery.
In slightly more detail, a superposition is described as many possible positions of the system. The different positions are indexed by complex numbers—that is, every position has a complex number associated with it. When a superposition collapses, the chances that it collapses to a particular position are determined by that complex number.
The measurement does not tell you what was there before. Rather, the measurement produces the outcome.
This is crazy! We have just proved that objects cannot have certain properties until we measure them.
The answer is that each of the two particles is in a superposition of spinning both ways. Only when one of the particles is measured does it randomly collapse into a particular spin direction. And here is the amazing part: the instant one of the particles collapses one way, the other must collapse the exact opposite way. This is true even if the two particles are light-years apart. That is, in order for the universe to maintain conservation of spin, measuring one particle’s spin will collapse the other particle’s spin across the universe. Although these two particles are far away, they are entangled with each other. How can this happen?
In 1964, Bell published a paper, “On the Einstein-Podolsky-Rosen Paradox,” which famously showed that no regular hidden variables can explain away the mysteries of quantum entanglement.
When Ann does a measurement of her particle, Bob’s particle collapses from its superposition to have the opposite spin of Ann’s measured spin. This means that the usual notion of space that we have is wrong: measurements do affect distant objects.
Entanglement shows that there are no closed systems. Every part of a system can be entangled with other parts outside of the system. All different systems are interconnected and the whole universe is one system. One cannot understand a system without looking at the whole universe. That is, “the whole is more than just the sum of its parts.”
This again conforms to the Wholeness Postulate. Here we see that the outcome depends on the setup of the entire experiment,
They found that particles had discrete spinning states, space was discrete, and time was also discrete. Electrons jump from shell to shell but do not cover the intermediate distance. Such jumps are called “quantum leaps.”
• Properties of objects have more than one value at a time. • There is no way to determine which value will be observed when the property is measured. • There are pairs of properties for which there is an inherent limitation of our ability to know their values. • Reality depends on how it is measured. • Distant parts of our universe are strangely interconnected. • Experimenters and their free will cannot be separated from their experiments.
The worst part of the Copenhagen interpretation is that there is a feeling of unquestioned dogma: “This is the way it is and one should not ask about it.” There is something unsatisfying when scientists tell you there is no real explanation for phenomena and the question does not make sense.
All these different choices are essentially ideas that make you feel good about the universe you live in. Unless someone comes up with some experiment that can show one view to be correct and the others to be false, there is no scientific reason to choose any of them. For now, the correct interpretation of quantum mechanics is beyond science.
The central idea of relativity theory is that properties of the physical universe depend on how they are measured. There are no absolute measurements. This is in sharp contrast to our naive notions about the universe.
David Hume showed that there is no logical reason why induction should work.
Induction is going from observing many single instances to a general rule. Going in the opposite direction—from a general rule to a conclusion about a particular instance—is called deduction.
More importantly, for our purposes, what it shows is that what we think we know about the universe is not necessarily true. It simply has not been falsified yet. While
For Popper, all scientific knowledge is provisional and not absolute.
Kuhn came to some of these ideas by studying the works of Aristotle. He realized that from the point of view of a Newtonian scholar, Aristotle is totally wrong and a bad physicist.
“Every answer given on principles of experience begets a fresh question, which likewise requires its answer and thereby clearly shows the insufficiency of all physical modes of explanation to satisfy reason.”
Philosophy is written in that great book which continually lies open before us (I mean the Universe). But one cannot understand this book until one has learned to understand the language and to know the letters in which it is written. It is written in the language of mathematics, and the letters are triangles, circles and other geometric figures. Without these means it is impossible for mankind to understand a single word; without these means there is only vain stumbling in a dark labyrinth.
very strange complex numbers in an essential way. It turns out that the superposition, which is at the core of quantum theory, is described by these complex numbers. More specifically, the many positions of a quantum state are indexed by complex numbers. Those strange curiosities are needed to describe our world.
By incorporating all these findings, science has come to accept something termed the principle of mediocrity or the Copernican principle that says that what we observe in the universe is not special in any sense. We are a typical species on an ordinary planet spinning around a commonplace sun in a usual galaxy. There is nothing special about us.
As we showed, there are countably infinite statements that are also true but unprovable in Peano Arithmetic. It would be nice to see more such statements that are “natural” and look like “real mathematics.” One such statement is Goodstein’s theorem.
“hereditary base-2 notation.”
Goodstein proved this amazing theorem using infinitary methods and the full power of set theory. In 1982, Laurie Kirby and Jeff Paris demonstrated that this theorem can only be proved using such infinitary methods and that although the theorem can be stated in the language of Peano Arithmetic, it cannot be proved in that system. The numbers simply get too big for that system. So Goodstein’s theorem, like Gödel’s sentence, is true but unprovable in Peano Arithmetic.
any self-referential system, no matter how powerful, is somewhat limited.
Nevertheless, it is unnerving that proving the consistency of the logical systems that are at the foundation of mathematics and science is beyond the bounds of reason.
• Gödel’s first incompleteness theorem says that certain statements are true but unprovable in some finite arithmetical system. But we saw that the same statements are provable in stronger systems.
Gödel’s second incompleteness theorem says that the consistency of a finite arithmetical system is unprovable within that system. Gentzen showed that the consistency is provable in the stronger system of ZFC.
The human brain is probably the most complicated machine in the entire universe and we are hundreds of years away from actually understanding how the human brain works.
The eternal silence of these infinite spaces fills me with dread.
Scientists are not outside the universe looking in. Rather, they are part of the universe and trying to make sense of it. They are part of the phenomena they are studying.
That is, the universe is the ultimate self-referential system: the universe uses scientists to study itself.
Many of the limitations that we found were simply byproducts of self-reference. Once a system has the ability to talk about itself and deal with its own properties, there will be limitations of the system.
The time we spend with our loved ones is treasured even though there is no logical necessity for it. We feel pain when we are distant from our loved ones. Our decisions are not made on the basis of logic and reason. Instead we use aesthetics, practical experience, moral inclinations, gut impulses, emotions, intuitions, and feelings. In this sense, every one of us already transcends the bounds of reason.