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Shawn Clayton
Shawn Clayton

The Unknowable Universe: How Science, Mathematics, and Logic Fail to Explain Everything



The Outer Limits of Reason: What Science, Mathematics, and Logic Cannot Tell Us




Have you ever wondered about the limits of human knowledge? Is there anything that we cannot know, understand, or explain? Are there some questions that are beyond the reach of science, mathematics, and logic? If so, what are they and why are they so hard?




the outer limits of reason epub 11


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In this article, we will explore some of the fascinating topics covered in the book The Outer Limits of Reason by Noson S. Yanofsky. This book investigates what cannot be known, rather than what is known, about the universe. It studies what science, mathematics, and logic tell us cannot be revealed, predicted, described, or proven. It challenges our deep-seated beliefs about our universe, our rationality, and ourselves.


We will look at some of the examples of unsolvable problems, paradoxes, and limitations that Yanofsky discusses in his book. We will see how language, philosophy, infinity, computing, physics, and mathematics can lead us to surprising and sometimes disturbing conclusions. We will also see how these topics reveal the structure and limitations of reason itself.


Language Paradoxes




Language is one of the most powerful tools that humans have. We use language to communicate, express ourselves, learn new things, create art, and much more. Language is also the basis of logic and reasoning. However, language can also be used to create nonsensical sentences and paradoxical statements. These are sentences or statements that either have no meaning or contradict themselves.


Philosophical Conundrums




Philosophy is the study of the fundamental nature of reality, knowledge, and existence. Philosophy can help us to examine our assumptions, clarify our concepts, and challenge our common sense. However, philosophy can also lead us to conundrums that seem to defy logic and rationality. These are problems or puzzles that have no clear or satisfactory solution.


The Liar Paradox




One of the most famous paradoxes in philosophy is the liar paradox. This paradox involves a statement that refers to itself and contradicts itself. For example, consider the statement:



This statement is false.


Is this statement true or false? If it is true, then it is false, as it says. But if it is false, then it is true, as it says. Either way, we get a contradiction. This paradox shows that self-referential statements can be problematic and lead to logical inconsistencies.


The Sorites Paradox




Another paradox in philosophy is the sorites paradox. This paradox involves a vague statement that leads to a heap of problems. For example, consider the statement:



A heap of sand has at least 1000 grains of sand.


This statement seems reasonable enough. But what if we remove one grain of sand from the heap? Is it still a heap? What if we remove another grain? And another? At what point does the heap stop being a heap? There seems to be no clear answer. This paradox shows that vague statements can be problematic and lead to borderline cases.


Infinity Puzzles




Infinity is one of the most fascinating and mysterious concepts in mathematics. Infinity can be used to describe something that has no end, no limit, or no boundary. Infinity can also be used to create mind-boggling scenarios and paradoxes. These are situations or arguments that involve infinite quantities or processes.


Different Levels of Infinity




One of the most surprising discoveries in mathematics is that there are different levels of infinity. Some infinities are bigger than others, and some infinities are uncountable. This means that there are some sets of things that have more elements than others, even though they are both infinite. And there are some sets of things that cannot be matched one-to-one with the natural numbers, even though they are both infinite.


Cantor's Diagonal Argument




One of the proofs that shows that there are different levels of infinity is Cantor's diagonal argument. This proof shows that there are more real numbers than natural numbers. The real numbers are the numbers that can have decimals, such as 3.14 or 0.333... The natural numbers are the counting numbers, such as 1, 2, 3, and so on.


The proof works by assuming that there is a one-to-one correspondence between the real numbers and the natural numbers. That is, we assume that we can list all the real numbers in a table like this:



10.123456789...


20.987654321...


30.314159265...


40.271828182...


50.141421356...


60.693147180...


70.577215664...


80.414213562...


90.301029995...


100.176091259...


......


We then construct a new real number by taking the diagonal digits of this table and changing them in some way. For example, we can add 1 to each digit (and wrap around 9 to 0). This gives us a new real number like this:



0.234768091...


Hilbert's Hotel




Another example that shows how infinite sets can behave strangely is Hilbert's hotel. This is a thought experiment that involves a hotel with infinitely many rooms, all of which are occupied. The hotel can still accommodate new guests, even infinitely many of them, by shifting the existing guests to different rooms.


For example, if one new guest arrives, the hotel can move the guest in room 1 to room 2, the guest in room 2 to room 3, and so on. This frees up room 1 for the new guest. If infinitely many new guests arrive, the hotel can move the guest in room 1 to room 2, the guest in room 2 to room 4, the guest in room 3 to room 6, and so on. This frees up all the odd-numbered rooms for the new guests.


This shows that adding or subtracting infinite quantities does not change the size of an infinite set. This is different from how finite sets work. For example, if a hotel with 100 rooms is full, it cannot accommodate any new guests, no matter how they rearrange the existing guests.


Computing Complexities




Computers are one of the most powerful tools that humans have. We use computers to perform complex calculations, store and process large amounts of data, communicate and share information, and much more. Computers are also the basis of artificial intelligence and machine learning. However, computers have their own limitations and challenges. There are some problems that are too hard for computers to solve, or even impossible.


The Halting Problem




One of the most famous problems that shows that there are some questions that computers cannot answer is the halting problem. This problem asks whether there is a general way to determine if any computer program will eventually stop or run forever. For example, consider a program that prints out all the prime numbers. Will this program ever stop? The answer is no, because there are infinitely many prime numbers.


Turing Machines




To formalize the halting problem, we need a simple model of computation that can simulate any computer program. Such a model is called a Turing machine. A Turing machine consists of a tape that can store symbols, a head that can read and write symbols on the tape, and a set of rules that tell the head what to do based on the current symbol and state.


For example, here is a Turing machine that adds one to a binary number:



StateSymbolNew StateNew SymbolMove


Start0Start0Right


Start1Start1Right


StartBAddBLeft


Add1Add0Left


AddBHalt1-


This Turing machine starts by moving to the right until it reaches a blank symbol (B). Then it moves to the left and changes the last symbol to its opposite. If the last symbol was 0, it changes it to 1 and halts. If the last symbol was 1, it changes it to 0 and continues to the left. If it reaches another blank symbol, it writes a 1 and halts. For example, if the tape initially contains 1011, the Turing machine will change it to 1100.


Undecidable Problems




The halting problem asks whether there is a Turing machine that can take any other Turing machine and its input as input, and output yes or no depending on whether the given Turing machine halts or not on the given input. For example, if we give the Turing machine that adds one to a binary number and the input 1011 as input, the halting Turing machine should output yes, because the adding Turing machine halts on that input.


The answer to the halting problem is no. There is no such Turing machine that can solve the halting problem for all possible inputs. This can be proven by a contradiction. Suppose there is such a Turing machine H that can solve the halting problem. Then we can construct another Turing machine D that takes a Turing machine M as input, and does the following:



  • Run H on M and M as input.



  • If H outputs yes, then D goes into an infinite loop.



  • If H outputs no, then D halts.



Now what happens if we give D itself as input? If D halts on D, then H must output no on D and D. But this means that D does not halt on D, which is a contradiction. If D does not halt on D, then H must output yes on D and D. But this means that D goes into an infinite loop on D, which is also a contradiction. Either way, we get a contradiction. Therefore, there is no such Turing machine H that can solve the halting problem.


This shows that the halting problem is undecidable. There is no algorithm or computer program that can answer it for all possible inputs. There are many other problems that are undecidable as well, such as whether two Turing machines are equivalent, whether a given mathematical statement is provable, or whether a given computer program has bugs.


Scientific Limitations




Science is one of the most successful endeavors of human civilization. Science can reveal the mysteries of nature and explain how things work. Science can also make predictions and test them with experiments and observations. Science can also help us to solve practical problems and improve our lives. However, science has its own limitations and challenges. There are some things that science cannot tell us, or that are beyond the scope of scientific inquiry.


The Quantum World




One of the most fascinating and puzzling areas of science is quantum mechanics. Quantum mechanics describes the behavior of subatomic particles, such as electrons, photons, and quarks. Quantum mechanics challenges our intuition and common sense in many ways. For example, quantum mechanics shows that:



  • Subatomic particles can exist in superpositions of two or more states at once, until they are measured.



  • Subatomic particles can be entangled with each other, so that their states are correlated even when they are far apart.



  • Subatomic particles can tunnel through barriers that they should not be able to cross.



  • Subatomic particles can have properties that are uncertain or indeterminate until they are measured.



These phenomena have been confirmed by many experiments and have many applications in technology, such as lasers, transistors, and quantum computers. However, they also raise many questions and paradoxes about the nature of reality, causality, and information.


Heisenberg's Uncertainty Principle




One of the principles that states that there is a limit to how precisely we can measure certain properties of a quantum system is Heisenberg's uncertainty principle. This principle states that there is a trade-off between the accuracy of measuring the position and the momentum of a particle. The more precisely we measure one, the less precisely we can measure the other. For example, if we measure the position of an electron very accurately, we will have no idea about its momentum, and vice versa.


This is not because of any limitations of our instruments or methods, but because of the inherent nature of quantum systems. The position and momentum of a particle are not well-defined until they are measured. They are subject to quantum fluctuations that make them uncertain and indeterminate. This means that there is a fundamental limit to how much we can know about a quantum system.


Schrödinger's Cat




One of the thought experiments that illustrates the paradoxical nature of quantum superposition and measurement is Schrödinger's cat. This experiment involves a cat that is placed in a sealed box with a radioactive atom, a Geiger counter, a hammer, and a vial of poison. The radioactive atom has a 50% chance of decaying in one hour. If it decays, the Geiger counter detects it and triggers the hammer to break the vial of poison, killing the cat. If it does not decay, nothing happens and the cat remains alive.


According to quantum mechanics, before we open the box and observe the cat, the radioactive atom is in a superposition of two states: decayed and not decayed. This means that the cat is also in a superposition of two states: dead and alive. The cat is neither dead nor alive, but both at once, until we open the box and collapse the superposition into one definite state. This seems absurd and contradictory to our common sense.


There are many interpretations and debates about what this experiment means and what happens when we open the box. Some say that there are parallel universes where both outcomes happen. Some say that the cat is always either dead or alive, but we do not know which until we look. Some say that the cat is an observer itself and collapses its own superposition. Some say that there is no objective reality at all, but only subjective experiences.


Conclusion




In this article, we have explored some of the topics covered in the book The Outer Limits of Reason by Noson S. Yanofsky. We have seen some examples of unsolvable problems, paradoxes, and limitations that arise from language, philosophy, infinity, computing, physics, and mathematics. We have also seen how these topics reveal the structure and limitations of reason itself.


We have learned that there are some things that we cannot know, understand, or explain, and some questions that are beyond the reach of science, mathematics, and logic. However, this does not mean that we should give up on our quest for knowledge and understanding. Rather, it means that we should be humble and curious about our universe, our rationality, and ourselves.


FAQs





  • What is The Outer Limits of Reason?



  • What are some examples of language paradoxes?



  • What are some examples of infinity puzzles?



  • What are some examples of computing complexities?



  • What are some examples of scientific limitations?



What is The Outer Limits of Reason?




What are some examples of language paradoxes?




Language paradoxes are sentences or statements that either have no meaning or contradict themselves. Some examples are:



  • This statement is false.



  • I am lying.



  • The following sentence is true. The previous sentence is false.



  • This sentence has five words.



  • All generalizations are false.



What are some examples of infinity puzzles?




Infinity puzzles are situations or arguments that involve infinite quantities or processes. Some examples are:



  • There are more real numbers than natural numbers, even though they are both infinite.



  • A hotel with infinitely many rooms can accommodate infinitely many new guests, even if it is full.



  • There are infinitely many prime numbers, but only one even prime number.



  • A monkey typing randomly on a keyboard will eventually produce the complete works of Shakespeare, given enough time.



  • There is no largest natural number, but there is a smallest natural number.



What are some examples of computing complexities?




Computing complexities are problems that are too hard or impossible for computers to solve. Some examples are:



  • The halting problem: There is no general way to determine if any computer program will eventually stop or run forever.



  • The traveling salesman problem: There is no efficient way to find the shortest route that visits a given set of cities and returns to the starting point.



  • The encryption problem: There is no easy way to break a code that is based on a large prime number, unless you know the prime number.



  • The chess problem: There is no optimal strategy to win every game of chess, unless you can calculate all the possible moves and outcomes.



  • The artificial intelligence problem: There is no clear definition or test of what constitutes intelligence or consciousness in a machine.



What are some examples of scientific limitations?




Scientific limitations are things that science cannot tell us, or that are beyond the scope of scientific inquiry. Some examples are:



  • The quantum world: Quantum mechanics describes the behavior of subatomic particles, but it also challenges our intuition and common sense in many ways.



  • The relativity theory: Relativity theory describes the effects of gravity and motion on space and time, but it also leads to paradoxes and contradictions with classical physics.



  • The origin of the universe: The big bang theory explains how the universe began from a singularity, but it does not explain what caused the singularity or what happened before it.



  • The nature of life: Biology explains how living organisms function and evolve, but it does not explain what makes something alive or how life originated on Earth.



  • The meaning of existence: Science can explain how things work, but it cannot answer why they exist or what is their purpose.



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