Letters At The End Of A Math Proof Crossword
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An ematic proof is a finite set of ematic statements that form a valid and convincing argument for a desired conclusion based on the given assumptions.
Letters At The End Of A Math Proof Crossword
Why “convincing”? What does this mean? Unfortunately, the text doesn’t comment much on this. It only says that “convincing” is a psychological concept, so it is unsatisfactory from the point of view of formal evidence.
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This makes sense, of course, but I’m troubled by the notion that ematic evidence should be compelling. This seems to suggest that logically valid ematic evidence can somehow be rejected because it was “unconvincing”?
If there was only one ematic in the world, would he be able to produce an ematic proof?
The primary reason for writing a certificate is to communicate with other ematics. If empaths familiar with the relevant literature don’t understand your argument (ie don’t find it convincing), then I think the author has failed to write a proof.
Everything about ematic evidence is psychological. Including the valid and persuasive part. Who decides if it is valid and persuasive? The primary audience for the proof is the author of the proof. Presumably the author finds it valid and persuasive. If the evidence is convincing enough, others may agree.
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Understand that what is valid and persuasive depends on time and place. The geometric proofs of the ancient ematics were considered valid and convincing enough at the time, but later ematics have found that those proofs were not precise enough. For a nice discussion of this, read MSE question 80930 “What is the modern axiomatization of (Euclidean) plane geometry?”.
It becomes communication and cooperation between the author and the reader of the proof. Proof in itself has no power to convince a reader who does not want to be convinced, no matter how rigorous or valid the proof. The certificate can be rejected for various reasons. There are many examples of this in the history of ematics. A good example is the controversy over Cantor’s set theory.
It seems obvious to me that M, the ematic, could do whatever M wanted, and no one else could have any ematic objections. If M is convinced of the validity of an ematic proof, then that is all that is required of a proof. After all, who decides whether any evidence is “false” if not the mathematician M? Of course, M can spot an error in the certificate and correct it. Thus, validity depends on time. The whole idea of ”evidence” is an artifact of history. What is the justification and meaning of writing and reading proofs without context?
There is often, if not always, some gap in emetic evidence, and thus we have the “convincing” part of the definition. Note that the author shows gaps in his proof when he talks about the rule that $a=b$ and $b=c$ we can deduce $a=c$. I would take such fundamentals of equation reasoning as a presupposition for some arithmetical foundations, but I do not consider the partitioning of natural numbers into “even” and “odd” natural numbers as presupposed. So, personally, it doesn’t bother me that evidence is dismissed if it is found to be “inconclusive”.
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, because one would believe that there is a third class of natural numbers that is neither odd nor even, and some members of this class have even squares. Or someone may not have shown that such a partition exists for the arithmetic at hand. As a proof, I therefore do not consider it convincing, unless I missed some point in the text where the author first stated that every natural number is exclusively either odd or even.
When I said I don’t find the evidence compelling, I don’t just mean that the evidence isn’t formulated. When I said that I did not find the proof convincing, I mean that if the proof were formalized, it would require non-logical concepts that the text does not imply beyond what it already contains. Although I don’t think dividing natural numbers into odd and even numbers is by any means impossible, it is not purely due to logic or equation reasoning, but is partly related to the nature of natural numbers. The reasoning of a proof can be considered valid in the sense that it reaches the correct conclusion and no step goes from the right step to the wrong step, but the reasoning is insufficient because it does not provide enough detail to conclude it conclusively. claim. More precisely, it is correct to say that “if a natural number $n$ cannot be odd, then $n$ must be even”, but I don’t think it consists of an arithmetic axiom of natural numbers. As far as I can tell, the proof needs the concept of even and odd numbers to form a section of the natural numbers, and the author has not proved it or even written it as a necessary lemma for the proof. So there is an illogical gap in the proof, and if one did not see how this gap could be filled, or believes that filling the gap is a serious challenge, then rejecting the proof seems perfectly reasonable.
Sometimes ematics also have different requirements for evidence. Have you heard of intuitionistic logic or constructivist ematics? Some ematics do not allow the law of the excluded middle to be used in ematic proofs. So they can reject certain evidence.
An ematic proof is a finite set of ematic propositions that form a valid argument for a desired conclusion from the given assumptions.
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If another person reads and accepts that your set of ematic claims is valid, and thus accepts that it is proof according to the definition above, then it can be argued that you have created a “convincing” proof, thereby including or omitting “convincing” from the definition of this scenario wouldn’t matter.
Suppose, however, that you cannot find another person who agrees that your set of ematic claims is valid. The key question here, then, is whether a “valid set of claims” is objective fact or social convention. If the latter, then it is impossible to have a “valid” but “unconvincing” set of propositions (since the set of propositions would only be “valid” if there is a social contract, which requires “convincing”). In this case, adding the word “persuasive” to the definition would not affect whether or not the set of statements would be classified as evidence. However, if the validity of an argument is an objective property that exists independently of humans, then one could argue that an unconvincing but nonetheless valid argument could indeed exist, which I think would merit the label “evidence”.
Therefore, I have shown that depending on your philosophy of ematics, either (1) adding the word “convincing” to a definition does not change the content of the definition, or (2) there can be evidence that is not “convincing”. As such, regardless of whether clause (1) or (2) is used, the word “convincing” can be safely removed from the definition of evidence.
No, you don’t want to sell anything. No need to convince anyone. Proofs that leave the reader dazzled and scratching their heads, after the robot has followed and validated every step, are the best, IMO. But I guess that’s a matter of taste.
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(It just has to be right. For some people, that’s convincing enough. But when it’s not, that’s where the fun begins.)
Perhaps the wording of that quote is a little less… strict than it could be. The “validity” of the evidence matters; “Compelling” is secondary. But it must be convincing (the reader can follow the steps and see that they seem to lead logically to the conclusion) for other ematicians to understand that the proof is valid.
Ideally, you just want to verify deterministically that the proof shows unambiguously that a given conclusion follows from some set of accepted axioms. This is the theory of ematic proof, but most proofs are not precise enough to achieve this without the possibility of human error.
In computers, automatic proof of sentences offers a higher level. But it is also significantly more complicated, and relatively little evidence is verified in this way. Basically, the problem is that automatically checking things this way, the certificate can’t skip any steps, even the “obvious” ones.
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So when a human writes a proof for human consumption, they have to convince the reader that each step is logically valid, even if the computer isn’t convinced.
Only the simplest proofs are written down step by step, completely formal, so that a computer program could mechanically check each line. For example, no one writes proofs that if $x = y + 3 $, $x – y = 3 $, from the axioms. If someone