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Thread: How a wrong logic could affect mathematics?

  1. Top | #11
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    Logic can never be wrong as logic when the rules are applied properly. As the saying goes in software, GIGO garbage in garbage out. Software always works exactly as it is coded.

    Just because a syllogism is logically valid, conclusion follows from premise, does not mean the syllogism was properly constructed to solve a problem. GIGO again.

    One of the consequences of the Incompleteness Theorem is that in an axiomatic system there are truths that can not be proven in the system. Euclidean geometry is an example. A point is infinitely small and massless, a line is comprised of an infinite number of points. Not provable in geometry.

    It has consequences. I write a program. A write a second program to validat e the first program. How do I validate the second program?

    Logic is the same. You can not use logic as an absolute proof of logic. What you demon stare is whiter or logical statements are valid within the system. Whether there can be a hidden flaw is not provable. That was in part the genesis of the Turing Machine.

    In math there is the Laplace and Fourier Transform's. It is everywhere in science and engineering. It rests on the idea that there is for any given function there is only one unique pair of transforms. There is a proof that says there is only one unique transform pair for any function. If an exception were found it could have serious consequences.

    How do you prove the proof is correct and without flaw? The truth table of an AND function is axiomatic, it is not logically provable.

  2. Top | #12
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    Quote Originally Posted by Speakpigeon View Post
    I believe mathematical logic is wrong. I mean, really, really wrong. I mean actually all wrong. However, I'm only really worrying, and more generally, as to the possible consequences for mathematics of using a formal logic that would be wrong.

    This question is in fact quite difficult to assess. Nearly all mathematicians use in fact their logical intuition to prove theorems. Thus, they don't have to rely on any method of formal proof and thus it doesn't seem to matter that mathematical logic should be wrong. At the same time, most mathematicians probably receive a comprehensive training in formal logic, and I can indeed routinely spot problematic statements, presented as "obviously" true, being made by mathematicians when they discuss formal logic questions, suggesting that their logical sense may be wrongly affected by their formal logic training. Yet, I'm not sure whether that actually affects the proof mathematicians produce in their personal work.

    It seems to me it's inevitable that it does. I know of specific proofs that are wrong in the sense that it's not something humans would normally accept. Mathematicians who accept them are obviously affected by their training in formal logic. However, these are proofs of logical formulas, not of mathematical theorems and these are much more difficult to assess in this respect.

    Yet, even if it is the case that actual proofs done by mathematicians using their intuition are wrongly affected by their training in formal logic, I'm still not clear what could be the consequences of that in practical term.

    One possible method to assess the possible consequences would be to compare proofs obtained using different methods of mathematical logic, such as relevance logics, intuitionistic logics, paraconsistent logics etc. However, I can't find examples of mathematical theorems proved using these methods. Further, all these methods are weaker than standard, "classical", mathematical logic, meaning that they deem valid a smaller number of logical implications and therefore, presumably, would end up with a smaller subset of the theorems currently accepted by mathematicians. Which may be good or bad but how do we know which?
    EB
    You said two geometries conflict, which and how, link?

    Ig you read a book on proofs and did examples yu would understand.

    In Calculus we learn intergration by parts. There is a form but no rules or systematic merthods to apply it. It is part intuition from exerince and part trial and eror. There is no way for a given function can actualy be solved through intergrtionnby parts.

    It is the same with prrofs. There is no structured method of any kind that given a term a proof results.

    Like integratiin by parts it is part intuition from experience, knowledge of math, and trial and error.

    Computers are now being used to derive proofs. Millions of trial and error trials can be run quickly. A quote attributed to AE, the greatest tool a scientist gas is a wastebasket.

    So in this case you are right, mathematicians can make mistakes. Mistakes can exist for years before being corrected. But that is not breaking or profound news or revelation. What is your point? We know that.

    The question as to whether there is a way to prove all theorems by an algorithm goes bc k at least to the early 20th century. The question was part of the genesis of the Turing Machine. A universal algorithmic processor. It is also related to the derivation of the Incompleteness Theorem. No consistent logical axiomatic system can be complete. Consistent in a general sense means that no matter how the system is applied, for example geometry or algebra, the answer will always be the same.

    I covered this stuff in a Theory Of Computation class. I read a book on doing proofs so I could read proofs for theorems in engineering text books. You may not think so, engineering foundations contains many proofs. A lot of the control systems theorems and proofs were developed by a mathematician-engineer Bode at Bell in the 1939s. Used to have a copy of his book.

    For me this all was not an academic debate.
    Last edited by steve_bank; 05-27-2019 at 04:55 AM.

  3. Top | #13
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    I asked the question because I knew you would not understand the question.

    What does A AND B mean? It does not mean anything. It is a definition, defined by a truth table.

    Given the logical definitions such as AND, OR, and the rest can it be logically proven that combinations of functions in formal logic will never lead to erroneous results? If so how?

  4. Top | #14
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    Quote Originally Posted by Speakpigeon View Post
    I believe mathematical logic is wrong. I mean, really, really wrong. I mean actually all wrong. However, I'm only really worrying, and more generally, as to the possible consequences for mathematics of using a formal logic that would be wrong.

    This question is in fact quite difficult to assess. Nearly all mathematicians use in fact their logical intuition to prove theorems. Thus, they don't have to rely on any method of formal proof and thus it doesn't seem to matter that mathematical logic should be wrong. At the same time, most mathematicians probably receive a comprehensive training in formal logic, and I can indeed routinely spot problematic statements, presented as "obviously" true, being made by mathematicians when they discuss formal logic questions, suggesting that their logical sense may be wrongly affected by their formal logic training. Yet, I'm not sure whether that actually affects the proof mathematicians produce in their personal work.

    It seems to me it's inevitable that it does. I know of specific proofs that are wrong in the sense that it's not something humans would normally accept. Mathematicians who accept them are obviously affected by their training in formal logic. However, these are proofs of logical formulas, not of mathematical theorems and these are much more difficult to assess in this respect.

    Yet, even if it is the case that actual proofs done by mathematicians using their intuition are wrongly affected by their training in formal logic, I'm still not clear what could be the consequences of that in practical term.

    One possible method to assess the possible consequences would be to compare proofs obtained using different methods of mathematical logic, such as relevance logics, intuitionistic logics, paraconsistent logics etc. However, I can't find examples of mathematical theorems proved using these methods. Further, all these methods are weaker than standard, "classical", mathematical logic, meaning that they deem valid a smaller number of logical implications and therefore, presumably, would end up with a smaller subset of the theorems currently accepted by mathematicians. Which may be good or bad but how do we know which?
    EB

  5. Top | #15
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    Quote Originally Posted by steve_bank View Post
    I asked the question because I knew you would not understand the question.

    What does A AND B mean? It does not mean anything. It is a definition, defined by a truth table.

    Given the logical definitions such as AND, OR, and the rest can it be logically proven that combinations of functions in formal logic will never lead to erroneous results? If so how?
    Quote Originally Posted by Speakpigeon View Post
    Now, please explain to us how the implication works... Nobody did, you know. So if you can do it, please show the world.
    EB
    Go on, do that.
    EB

  6. Top | #16
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    There is something about the generally accepted definition of “valid” that I can’t quite shake. It initially struck me as a definition with too broad a scope, allowing in things it shouldn’t (like contradictions) while capturing what it should (being collectively exhaustive one layer down), meaning what would otherwise be sound should the argument also have true premises. It’s an overreach as far as definitions go, so it seems more of a conflation with an off-target insight than anything that resembles the ‘is’ of identity.

    Apparently, according to mathematical logic, anything follows from a contradiction; it’s a good thing it doesn’t matter if that’s true since it’s guarentees any such argument it’s used in is unsound; but we really should consider a forward approach (what’s included) rather than a backwards approach (everything not excluded) when tweaking the definition.

    This is a stipulative definition for a reason. The surrounding construct is already in place. A deductive argument MUST be sound if it’s VALID and all premises are TRUE. The fact that a contradiction implies validity according to the definition of “validity” used is as about as meaningful as if were instead true that contradictions implies they’re not valid. That should mean something about whether or not the truth of whether contradictions imply anything is true is merely a function of satisfying a definition that is exclusionary (backwards).

    I’m sorry guys. I just don’t have the right words. That has to sound muddled. It’s just that an implication through form makes sense when parts of the conclusion is to be found in the premises; it’s ONLY through definition that allows for contradictions to imply anything. Had the definition been equally as flawed, but from an opposite direction, our reasoning that contradictions imply (not nothing) but that nothing is true would still be based on the satisfaction of a definition.

    Again, it’s not lexical but stipulative for a reason, and it’s slightly off target. It accomplishes our goal but is slightly too broad in that it captures and says of contradictions something that simply oughtn’t be considered as true. What should be the case is that contradictions within deductive arguments fail to imply anything; I’m not sure if contradictions should render an argument invalid, but it should definitely not be valid, (considering the distinction between not valid and invalid should it be important).

    I’m not sure how to reword the definition, but at the very least I could piece meal it, sloppy as that might be. Simply exclude contradictions within the definition.

  7. Top | #17
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    The way I look at it valid means consistent with rules. A valid sentence in terms of syntax and grammar. You can have a sentence that is correct garmmaticley and syntx wise, yet be completely meaningless.

    Lewis Carol's Jabberwocky poem.

    Semantics of course,

    In a discussion someone can have a valid argument, but it is not necessarily true. In a two sided debate an independent observer may say both sides have valid arguments. Given the hypotheses and premise a valid conclusion follows on both sides. The question being which of the hypothesis and premise is true in reality.

    A valid syllogism according the rules of logic does not necessarily mean the conclusion is true in reality, only that conclusion follows from premise.

    That is why pure linear logic is limited real world complex problems. All problems are not reducible to syllogisms..

  8. Top | #18
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    Quote Originally Posted by Speakpigeon View Post
    Quote Originally Posted by steve_bank View Post
    I asked the question because I knew you would not understand the question.

    What does A AND B mean? It does not mean anything. It is a definition, defined by a truth table.

    Given the logical definitions such as AND, OR, and the rest can it be logically proven that combinations of functions in formal logic will never lead to erroneous results? If so how?
    Quote Originally Posted by Speakpigeon View Post
    Now, please explain to us how the implication works... Nobody did, you know. So if you can do it, please show the world.


    EB
    Go on, do that.
    EB
    Asked and answered. A definition has no meaning, it is a 'definition'. Formal logic is a set of arbitrary rules based system.If it was not enough for you then try harder to understand. I can not connect the mental dots for you.

    You do not appear to actually have done much reasoning or work with logic and math. Try the short book on proofs I recommended and work some of the problems. It will all become a lot clearer.

    The questions you pose go a lot deeper. It touches on formal languages in Theory Of Computations and computer science. Computer science is the center for logic today. Formal logic in math is axiomatic, such as geometry. The system develops from a defined set of relations which are not necessarily provable. That is why I think the incompleteness Thermo applies. There is no way using logic to prove a particular set of logic will always be correct in a general sense.

    If you do not understand Gödel then you are thinking in pre 20th century philosophy. Aristotle thought given a set of basic principles the universe can be deuced logically. Gödel says not. In electrical engineering Gödel is known in general. I rerad a book about him.

    https://en.wikipedia.org/wiki/Axi

    An axiom or postulate is a statement that is taken to be true, to serve as a premise or starting point for further reasoning and arguments. The word comes from the Greek axíōma (ἀξίωμα) 'that which is thought worthy or fit' or 'that which commends itself as evident.'[1][2]

    The term has subtle differences in definition when used in the context of different fields of study. As defined in classic philosophy, an axiom is a statement that is so evident or well-established, that it is accepted without controversy or question.[3] As used in modern logic, an axiom is a premise or starting point for reasoning.[4]

    As used in mathematics, the term axiom is used in two related but distinguishable senses: "logical axioms" and "non-logical axioms". Logical axioms are usually statements that are taken to be true within the system of logic they define (e.g., (A and B) implies A), often shown in symbolic form, while non-logical axioms (e.g., a + b = b + a) are actually substantive assertions about the elements of the domain of a specific mathematical theory (such as arithmetic). When used in the latter sense, "axiom", "postulate", and "assumption" may be used interchangeably. In general, a non-logical axiom is not a self-evident truth, but rather a formal logical expression used in deduction to build a mathematical theory. To axiomatize a system of knowledge is to show that its claims can be derived from a small, well-understood set of sentences (the axioms). There are typically multiple ways to axiomatize a given mathematical domain.

    Any axiom is a statement that serves as a starting point from which other statements are logically derived. Whether it is meaningful (and, if so, what it means) for an axiom to be "true" is a subject of debate in the philosophy of mathematics

    Mathematical logic[edit]

    In the field of mathematical logic, a clear distinction is made between two notions of axioms: logical and non-logical (somewhat similar to the ancient distinction between "axioms" and "postulates" respectively).

    Logical axioms[edit]

    These are certain formulas in a formal language that are universally valid, that is, formulas that are satisfied by every assignment of values. Usually one takes as logical axioms at least some minimal set of tautologies that is sufficient for proving all tautologies in the language; in the case of predicate logic more logical axioms than that are required, in order to prove logical truths that are not tautologies in the strict sense.

    https://plus.maths.org/content/gouml...d-limits-logic

    Gödel proved that the mathematical methods in place since the time of Euclid (around 300 BC) were inadequate for discovering all that is true about the natural numbers. His discovery undercut the foundations on which mathematics had been built up to the 20th century, stimulated thinkers to seek alternatives and generated a lively philosophical debate about the nature of truth. Gödel's innovative techniques, which could readily be applied to algorithms for computations, also laid the foundation for modern computer science.

  9. Top | #19
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    So the answer is you don't have the beginning of a clue.
    EB

  10. Top | #20
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    Quote Originally Posted by fast View Post
    I’m not sure how to reword the definition, but at the very least I could piece meal it, sloppy as that might be. Simply exclude contradictions within the definition.
    The fact that their definition of validity is wrong shows mathematicians don't even understand logical validity. Mathematical logic is not even a restriction of logic. It's not logic at all.
    EB

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