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Post by Eugene 2.0 on Nov 12, 2021 11:03:26 GMT
Can logic be without axioms?
If we take x=x or ~{x≠x} as well as any other definitions of how our operators are being used, then we already introduce by it some axioms.
Math Has logic or Not?
If it has the logic, it has a metaphysical part. If it doesn't:
Let's say we're asking whether or not "+" is using mathematically. If it's being used mathematically then "+" must be placed somewhere at a formula. And it's presented must not be made deliberately (it has some logic). If "+" is being used without math, then we still have a question – is it still math? I think it must be clearly if only within math context "+" is "a sum" or anything that math does. Using plus without math doesn't make it be mathematically. In other words, only math context makes "+" be "+".). It results in the math as a whole, and any wholes are metaphysical.
Either way avoid dealing with axioms or with metaphysics doesn't seem to be optional.
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Post by joustos on Nov 12, 2021 15:31:01 GMT
Can logic be without axioms?==> By logic, do you mean "logical/rational thinking" or"the collectionof formulas of correct thinking" or a "deductive/formal system "[called also 'symbolic logic']? [While reading, I felt I was forgetting what Logic and Metaphysics are. My fault or yours? ... Difficult communication... If we take x=x or ~{x≠x} as well as any other definitions of how our operators are being used, then we already introduce by it some axioms. ==> For the sake of argument, take/assert x=x. Question: What axiom have you implicitly introduced? Or, I ask, is the assertion that x is equal to x the assertion of an axiom?Finally: it seems to me that what is being defined here is x, not the defining operator [namely =]. Incidentally, when speaking of a verbal definition, wherein concepts are at play, I would use the equivalence operator [namely = ] rather than the identity or tautology or mathematical operator [=]. Anyway, in A = B, what is being defined is A, not the operator. Math Has logic or Not?If it has the logic, it has a metaphysical part. If it doesn't: Let's say we're asking whether or not "+" is using mathematically. If it's being used mathematically then " +" must be placed somewhere at a formula. And it's presented must not be made deliberately (it has some logic). If "+" is being used without math, then we still have a question – is it still math? I think it must be clearly if only within math context "+" is "a sum" or anything that math does. Using plus without math doesn't make it be mathematically. In other words, only math context makes "+" be "+".). It results in the math as a whole, and any wholes are metaphysical. Either way avoid dealing with axioms or with metaphysics doesn't seem to be optional. |
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Post by Eugene 2.0 on Nov 12, 2021 15:54:57 GMT
Can logic be without axioms?==> By logic, do you mean "logical/rational thinking" or"the collectionof formulas of correct thinking" or a "deductive/formal system "[called also 'symbolic logic']? [While reading, I felt I was forgetting what Logic and Metaphysics are. My fault or yours? ... Difficult communication... If we take x=x or ~{x≠x} as well as any other definitions of how our operators are being used, then we already introduce by it some axioms. ==> For the sake of argument, take/assert x=x. Question: What axiom have you implicitly introduced? Or, I ask, is the assertion that x is equal to x the assertion of an axiom? Math Has logic or Not?If it has the logic, it has a metaphysical part. If it doesn't: Let's say we're asking whether or not "+" is using mathematically. If it's being used mathematically then " +" must be placed somewhere at a formula. And it's presented must not be made deliberately (it has some logic). If "+" is being used without math, then we still have a question – is it still math? I think it must be clearly if only within math context "+" is "a sum" or anything that math does. Using plus without math doesn't make it be mathematically. In other words, only math context makes "+" be "+".). It results in the math as a whole, and any wholes are metaphysical. Either way avoid dealing with axioms or with metaphysics doesn't seem to be optional. Can't say my knowledge of metaphysics is perfect. Even to students I try to explain it as about the area in philosophy that doesn't belong to the rest branches of the one. (Metaphysics is not epistemology, ethics, so on.) SEP or IEP defines it sharply better, but in my opinion I should try to express it by myself trying to express this notion using my previous experience of reading thinkers which uses metaphysics. Aristotle, as I presume, took it as the first deity or essence (can't say I used the correct translation here), involving into explanation of the primary matter or kinda. Quine (in "On What There Is") explained it as the subject of studying the more ambiguous questions in ontology and epistemology. In the topic I tried to use Metaphysics as something enough powerful to provide philosophical investigation. In other words, what if logic is optional? What else might be used – perhaps, the method of metaphysics, or the metaphysics itself. |
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Post by jonbain on Nov 12, 2021 20:49:01 GMT
Eugene 2.0The foundational axiom of any discussion is that it is possible to reach agreement on truth in the discussion ; or else there would not be any reason to have a discussion. But it is vital to see that 'pure math' cannot have any useful value. Math must always be applied. And its also vital to identify the context of that application very carefully. For example I can say that 1+1=3 because 1 man + 1 woman = a couple and a child. And that is not wrong; even though it appears illogical. So it is certainly not true in purely logical terms. So can we therefore conclude that the child existed before the man and woman met? Logic would say yes; though our limited empirical observation then appears to be wrong. This is why I like geometry as foundational. It is a truly empirical science that has no such paradoxes. And in fact; after studying the expansion of the universe - we then realize that there must be at least a fourth dimension of space; then we can see further evidence that the soul of that child must also have existed in extra dimensional space before the man and woman met. This free book is the most essential reading in terms of the philosophy of math: ned.ipac.caltech.edu/level5/Abbott/paper.pdf
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Post by Eugene 2.0 on Nov 12, 2021 22:42:57 GMT
Eugene 2.0The foundational axiom of any discussion is that it is possible to reach agreement on truth in the discussion ; or else there would not be any reason to have a discussion. But it is vital to see that 'pure math' cannot have any useful value. Math must always be applied. And its also vital to identify the context of that application very carefully. For example I can say that 1+1=3 because 1 man + 1 woman = a couple and a child. And that is not wrong; even though it appears illogical. So it is certainly not true in purely logical terms. So can we therefore conclude that the child existed before the man and woman met? Logic would say yes; though our limited empirical observation then appears to be wrong. This is why I like geometry as foundational. It is a truly empirical science that has no such paradoxes. And in fact; after studying the expansion of the universe - we then realize that there must be at least a fourth dimension of space; then we can see further evidence that the soul of that child must also have existed in extra dimensional space before the man and woman met. This free book is the most essential reading in terms of the philosophy of math: ned.ipac.caltech.edu/level5/Abbott/paper.pdf The agreement is a good and a reasonable purposes. It seems it sounds familiar. Recently I've discovered a Polish thinker named Quasimezh Ajdukiewicz [Quasi-mej I-do-kie-which], who tried to explain why the sense (e.g. a certain sentence has some sense) could be viewed, and by all means we were able to make some analysis of such. He gave the next example: if someone is shooting like he was injured, or got hit, then we might say that one suffered, and he felt pain. However, what if the symptoms of that one are absolutely the same as the symptoms of a hit or an injured person, and that person still is struggling to accept he has been injured, or he feels pain? What if that person refuses to call that symptom (or a series of them) "pain"? He might used to call this symptom differently, let's say, he called it "pleasure" instead of "pain". But in this case it's alright, and what do we need is to change the name from "the pain" to "the pleasure". What if such renaming won't help? It might have been possible – says Ajdukiewicz – but it would have not been this way. It would be impossible. Because this case shows no logic at all. We have either to accept illogical explanation, or to doubt in a patient's words. Anyway (sorry for a long begging to the commentary), Ajdukiewicz proposes to start from some agreement – as well as you've said. Frege, after he accepted his failure in his logicism (to construct the math using only logic), he tried to get Geometry as the one. He wasn't alone, David Gilbert did it also. I like Geometry. In school I got more A marks in Geometry, than in Algebra. My favourite ones were Geometry tasks. Algebra task were confusing – so, hell yeah, I do agree with what you have said. We have to get a clear agreement to see how Algebra is operating, while Geometry is quite clearer by itself. "The Wonderful Subject" – that's how I would call it. |
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Post by xxxxxxxxx on Nov 18, 2021 23:21:01 GMT
Can logic be without axioms?If we take x=x or ~{x≠x} as well as any other definitions of how our operators are being used, then we already introduce by it some axioms. Math Has logic or Not?If it has the logic, it has a metaphysical part. If it doesn't: Let's say we're asking whether or not "+" is using mathematically. If it's being used mathematically then " +" must be placed somewhere at a formula. And it's presented must not be made deliberately (it has some logic). If "+" is being used without math, then we still have a question – is it still math? I think it must be clearly if only within math context "+" is "a sum" or anything that math does. Using plus without math doesn't make it be mathematically. In other words, only math context makes "+" be "+".). It results in the math as a whole, and any wholes are metaphysical. Either way avoid dealing with axioms or with metaphysics doesn't seem to be optional. 1. x is viewed as variable, as a variable it is an axiom given "variability" is self evident. 2. To have logic without axioms is in itself an axiom of logic thus a contradiction occurs. 3. Counting is mathematics. Counting is the manifestation of forms as part of a set with this set being a form composed of forms. The manifestation of forms, through counting, necessitates math as requiring axioms given the forms are assumed as they are as forms. We accept forms for what they are. |
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Post by Eugene 2.0 on Nov 25, 2021 16:23:07 GMT
Can logic be without axioms?If we take x=x or ~{x≠x} as well as any other definitions of how our operators are being used, then we already introduce by it some axioms. Math Has logic or Not?If it has the logic, it has a metaphysical part. If it doesn't: Let's say we're asking whether or not "+" is using mathematically. If it's being used mathematically then " +" must be placed somewhere at a formula. And it's presented must not be made deliberately (it has some logic). If "+" is being used without math, then we still have a question – is it still math? I think it must be clearly if only within math context "+" is "a sum" or anything that math does. Using plus without math doesn't make it be mathematically. In other words, only math context makes "+" be "+".). It results in the math as a whole, and any wholes are metaphysical. Either way avoid dealing with axioms or with metaphysics doesn't seem to be optional. 1. x is viewed as variable, as a variable it is an axiom given "variability" is self evident. 2. To have logic without axioms is in itself an axiom of logic thus a contradiction occurs. 3. Counting is mathematics. Counting is the manifestation of forms as part of a set with this set being a form composed of forms. The manifestation of forms, through counting, necessitates math as requiring axioms given the forms are assumed as they are as forms. We accept forms for what they are. Yes, well said. |
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