Paradoxes In Mathematics
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A mathematical paradox is any statement (or a set of statements) that seems to contradict itself (or each other) while simultaneously seeming completely logical. Paradox (at least mathematical paradox) is only a wrong statement that seems right because of lack of essential logic or information or application of logic to a situation where it is not applicable. There are many paradoxes in mathematics.
The paradoxes of the philosopher Zeno, born approximately 490 BC in southern Italy, have puzzled mathematicians, scientists and philosophers for millennia. Although none of his work survives today, over 40 paradoxes are attributed to him which appeared in a book he wrote as a defense of the philosophies of his teacher Parmenides. Parmenides believed in monism, that reality was a single, constant, unchanging thing that he called 'Being'. In defending this radical belief, Zeno fashioned 40 arguments to show that change (motion) and plurality are impossible.
This is related to Russell's Paradox, which asked if the set of things that don't contain themselves contained itself. By creating self-destructing sets like these, Bertrand Russell and others showed the importance of establishing careful rules when creating sets, which would lay the groundwork for 20th-century mathematics.
The Interesting Number Paradox relies on an imprecise definition of \"interesting,\" making this a somewhat sillier version of some of the other paradoxes, like the heterological paradox, that rely on contradictory self-references.
Exploring more than seventy-five well-known paradoxes in mathematics, philosophy, physics, and the social sciences showing how reason and logic can dispel the illusion of contradiction.
The above results show an essential difference between abstract existence and trainability. Mathematically proving the existence of a good NN is not enough - one must also show that it can be obtained in practice. This paradox is very much related to the work of Alan Turing and Kurt Gödel. About 100 years ago, mathematicians set out to show that mathematics was the ultimate consistent language of the universe. There was a tremendous amount of optimism, similar to the optimism we see in AI today. However, Turing and Gödel turned this optimism on its head: it is impossible to prove whether certain mathematical statements are true or false, and some problems cannot be tackled with algorithms. Much later, the mathematician Steve Smale proposed a list of 18 unsolved mathematical problems for the 21st century. His 18th problem, featured in the title of our paper, concerned the limits of intelligence for both humans and machines. The mathematics of foundations, i.e., figuring out what is and is not possible, is now entering the world of AI.
The fastest human in the world, according to the Ancient Greek legend, was the heroine Atalanta. Although she was a famous huntress who even joined Jason and the Argonauts in the search for the golden fleece, she was renowned for her speed, as no one could defeat her in a fair footrace. But she was also the inspiration for the first of many similar paradoxes put forth by the ancient philosopher Zeno of Elea: about how motion, logically, should be impossible.
It might seem counterintuitive, but pure mathematics alone cannot provide a satisfactory solution to the paradox. The reason is simple: the paradox isn't simply about dividing a finite thing up into an infinite number of parts, but rather about the inherently physical concept of a rate.
The paradox identified by the researchers traces back to two 20th century mathematical giants: Alan Turing and Kurt Gödel. At the beginning of the 20th century, mathematicians attempted to justify mathematics as the ultimate consistent language of science. However, Turing and Gödel showed a paradox at the heart of mathematics: it is impossible to prove whether certain mathematical statements are true or false, and some computational problems cannot be tackled with algorithms. And, whenever a mathematical system is rich enough to describe the arithmetic we learn at school, it cannot prove its own consistency.
\"The paradox first identified by Turing and Gödel has now been brought forward into the world of AI by Smale and others,\" said co-author Dr. Matthew Colbrook from the Department of Applied Mathematics and Theoretical Physics. \"There are fundamental limits inherent in mathematics and, similarly, AI algorithms can't exist for certain problems.\"
\"When 20th-century mathematicians identified different paradoxes, they didn't stop studying mathematics. They just had to find new paths, because they understood the limitations,\" said Colbrook. \"For AI, it may be a case of changing paths or developing new ones to build systems that can solve problems in a trustworthy and transparent way, while understanding their limitations.\"
It's a monumental head-scratcher known as the 'grandfather paradox', but in September last year a physics student Germain Tobar, from the University of Queensland in Australia, said he has worked out how to \"square the numbers\" to make time travel viable without the paradoxes.
The new research smooths out the problem with another hypothesis, that time travel is possible but that time travellers would be restricted in what they did, to stop them creating a paradox. In this model, time travellers have the freedom to do whatever they want, but paradoxes are not possible.
Those two paradoxes are resolved in the same way. We came to a contradiction by assuming that a certain physical object or process exists. Once we abandon this assumption, we are free of the contradiction. These paradoxes show limitations of the physical world: a certain village cannot exist, or the actions of a time traveler are restricted. There are many similar examples of such physical paradoxes, and they are resolved in the same way.
So far we have seen that some paradoxes show us that there are physical objects or processes which cannot exist, while other paradoxes are about language and can be ignored. There is, however, a third class of paradoxes that come from language that cannot be ignored.
Mathematicians (and philosophers too) can use the fact that contradictions are not permitted to prove theorems. This is called reduction to the absurd or proof by contradiction: if you want to prove that a certain statement is true, assume it is false and derive a contradiction. Since there cannot be contradictions, the assumption of falsehood must be incorrect and therefore the original statement is true. Such proofs play a major role in modern mathematics.
Mathematical paradoxes are statements that run counter to one's intuition, sometimes in simple, playful ways, and sometimes in extremely esoteric and profound ways. It should perhaps come as no surprise that a field with as rich a history as mathematics should have many of them. They range from very simple, everyday common-sense issues, to advanced ones at the frontiers of mathematics. That is why this article has so many \"expertise level\" boxes above.
In addition to the seemingly paradoxical existence of non-Lebesgue-measurable sets, there are some more serious paradoxes. Perhaps the most famous is the Banach-Tarski paradox. This says that a sphere in 3 dimensions can be divided into a finite number of subsets (specifically, 5) which, without changing their shapes or sizes, can be reassembled into two spheres, each identical to the original one. This of course violates everything we thought we knew about what \"measure\" is supposed to mean. The way the paradox gets away with this trick is that the sets are not measurable.
Constructivist mathematicians (and some others) generally prefer proofs that do not require AC, on the grounds that such proofs are more \"clean\". However some proofs are significantly cleaner when the AC is assumed, providing elegance and generality for numerous theorems. The remarkable thing about the Axiom of Choice is that both accepting it and rejecting it lead to things that seem like paradoxes. Accepting AC is leads to the Banach-Tarski paradox in measure theory. Rejecting it turns out to break the notion of size in a different way, specifically without AC it isn't possible to prove that if two sets don't have the same number of elements then one of them must have fewer elements.
We could chart these competing ideas but, starting in the late 1920s, Kurt Gödel and subsequent researchers established the futility of seeking meaningful mathematical systems that are devoid of paradoxes. Indeed, one could claim that the power of mathematics is intrinsically tied in with its paradoxes.
Among mathematicians there are at least two tribes: those who seem not to dwell on internal inconsistencies and who are more interested in building new structures or discovering new truths, and those who would look more closely at the connections and contradictions that lie at the foundations of our mathematical methods. We need mathematicians who can be both, aiming to reach beyond existing horizons while still being reflective and critical about the models we use. We should not be paralysed by the paradoxes at the heart of our structures, but recognise that the paradoxes may be a source of power, and clarify and leverage them.
Note: learners who do well in Paradox will have typically taken at least a couple ofcollege-level classes in mathematics or computer science. On the other hand, Paradox doesnot presuppose familiarity with any particular branch of mathematics or computer science.You just need to feel comfortable in a mathematical setting.
Computer processing happens in digital electronic circuits. Theirdesign is based on Boolean expressions and the representation ofinteger numbers in the binary numeral system discovered byGottfried Leibniz (1646 - 1716). Instead of using ten digits as weare used to do in the decimal numeral system, the binary systemrequires only two. The sequence of integers starts with 0, followedby 1. Now there isn't a third digit to be used for 2, and so we mustoccupy another binary position, resulting in 10, followed by 11 (for3), and for 4, written as 100, we need yet another binary position.Arithmetic using this notation is based on extremely simple rules.Instead of having to learn a large number of single-digit sums youonly have to know that 0+0=0, 0+1=1+0=1 and 1+1=0, with 1 as carry.These simple formulæ have direct counterparts in Boolean expressions,and they are easy to represent by logic gates, devices implementingBoolean functions. As a consequence, all of the algorithms andmathematics that can be described with Boolean logic can be constructedusing logic gates. 59ce067264