How a wrong logic could affect mathematics?

[Angra Mainyu]

No. We know what is logic and we know what is science.

You claimed

These people don't even agree on what is logic and they don't even say what is logic. Read any textbook on logic and quote me where it explains what is logic. Not one ever does that nowadays.

My answer:

You could say the same about science, or morality, or metaphysics, or whatever. But you're just going on a tangent. I don't have time to address everything you claim, and on top of that defend myself against your misrepresentations (note: self -defense will always take priority over any discussion about logic, math, or whatever; consider that before attributing claims to me that I have not made or even suggested).

You could say the same, in that 'these people' (i.e.,scientists specialized in different fields, or philosophers specialized in different fields), sometimes (or often) disagree about the definition of their disciplines, and/or whether something is part of them.

What we won't know, even by reading logic textbooks, is what is mathematical logic. And yet, everybody behaves as if mathematical logic was something like the study of human reasoning. You yourself is so confused in your explanations I still don't know what your view is.

No, you are confused about my explanations, because even when I carefully debunk your claims, you still do not get it, and you keep misrepresenting my views despite detailed explanations. On the other hand, you refuse to give any arguments supporting the claims of yours I debunk. You just claim.

You reply but you never answer my questions.

That is false. It is you who refuse to provide arguments in support of your main claims. I, on the other hand, address your questions and debunk your claims (those that need debunking and I feel like debunking, that is). But regarding questions, the central question in this thread - indeed, the title question - is "How could a wrong logic affect mathematics?"

I already gave you my answer, and establish that it is true: The answer is that, under the hypothesis that the concept of validity used in mathematical logic (as provided by you in the OP) is wrong in the sense that it fails to match the usual concept (so, there are arguments that are valid in classical mathematical logic but aren't valid in human logic), and under the further hypothesis that every statement is either true or false (which is your belief, as you said here), mathematical logic is a tool superior to human logic as a means of finding mathematical truth.

Your failure to follow my arguments does not mean that I failed to answer your question. I did answer it, repeatedly, and gave a detail argument showing that my answer was true. The consequences of having the wrong logic are, in this case and under the hypotheses in question, that mathematics can do better with the wrong logic than it could with the right logic, as the wrong logic is a superior tool for finding mathematical truth than the right logic. Of course, here, 'wrong' and 'right' are assessed from the perspective of whether they match human logic, not whether they are conducive to finding mathematical truth.

To everybody else, to use an incorrect logic by definition will lead to invalid conclusions. Not to you apparently. You argued that these invalid conclusions will nonetheless be valid according to the notion of validity used in this incorrect logic. Alright. I've enough. Please ignore me from now on.

No, of course that is not what I argued. That vastly misrepresents my arguments.
First, it is trivially obvious that those invalid deductions would be valid according to the notion of validity used in this thread. I do not need to argue for that - not even here (well, I don't think I do, but there might be someone who doesn't see even something that trivial).
Second, what I argued for - and, indeed, what I successfully established - is that those invalid deductions would nonetheless reliably lead from truth to truth, and in fact, the definition of validity in the thread, while wrong by assumption (i.e., 'wrong' as in failing to capture human logic) would provide a superior tool for finding mathematical truths - superior to the correct definition, that is (readers: okay, that's trivial too, but there are different degrees of triviality.).

By the way, I do not put people on ignore, ever, for several reasons. The most important one (but not the only important one) is that I want to know if they attack me, and how they do so, so that I can fight back.

 

[A Toy Windmill]

Arguments with contradictory premises, or generally false premises, are easily formalised in Aristotelian logic.

You formalize into a formal logic. Aristotle had only one formal logic, his syllogisms, and no syllogism has contradictory premises. So arguments with contradictory premises are not formalizable in Aristotelian logic.

 

[steve_bank]

Saying logic is like saying philosophy or physucs. Ypu have to specify what area of logic you refer to and define it with context before debating.

A philosophical definition.

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

Logic (from the Ancient Greek: λογική, romanized: logikḗ[1]) is the systematic study of the form of valid inference, and the most general laws of truth.[2] A valid inference is one where there is a specific relation of logical support between the assumptions of the inference and its conclusion. In ordinary discourse, inferences may be signified by words such as therefore, thus, hence, ergo, and so on.

There is no universal agreement as to the exact scope and subject matter of logic (see § Rival conceptions, below), but it has traditionally included the classification of arguments, the systematic exposition of the 'logical form' common to all valid arguments, the study of proof and inference, including paradoxes and fallacies, and the study of syntax and semantics. Historically, logic has been studied in philosophy (since ancient times) and mathematics (since the mid-19th century), and recently logic has been studied in cognitive science (encompasses computer science, linguistics, philosophy and psychology).

In math, science, and engineering logic is an axiomatic system with postulates, axioms, and rules. As shown below logic in math is diverse. There is a lot of depth and make no pretense of having any expertise.

Mathematically A ^ B is a definition with a truth table

AB Evaluates To
FF F
FT f
TF F
TT F
https://en.wikipedia.org/wiki/Mathematical_logic

Mathematical logic is a subfield of mathematics exploring the applications of formal logic to mathematics. It bears close connections to metamathematics, the foundations of mathematics, and theoretical computer science.[1] The unifying themes in mathematical logic include the study of the expressive power of formal systems and the deductive power of formal proof systems.

Mathematical logic is often divided into the fields of set theory, model theory, recursion theory, and proof theory. These areas share basic results on logic, particularly first-order logic, and definability. In computer science (particularly in the ACM Classification) mathematical logic encompasses additional topics not detailed in this article; see Logic in computer science for those.

Since its inception, mathematical logic has both contributed to, and has been motivated by, the study of foundations of mathematics. This study began in the late 19th century with the development of axiomatic frameworks for geometry, arithmetic, and analysis. In the early 20th century it was shaped by David Hilbert's program to prove the consistency of foundational theories. Results of Kurt Gödel, Gerhard Gentzen, and others provided partial resolution to the program, and clarified the issues involved in proving consistency. Work in set theory showed that almost all ordinary mathematics can be formalized in terms of sets, although there are some theorems that cannot be proven in common axiom systems for set theory. Contemporary work in the foundations of mathematics often focuses on establishing which parts of mathematics can be formalized in particular formal systems (as in reverse mathematics) rather than trying to find theories in which all of mathematics can be developed.

I believe the underlined addresses you question. Are you refuting math over the last 200 years?

 

[Speakpigeon]

Again specific examples and the context.

They are publicly available.
EB

 

[Speakpigeon]

Arguments with contradictory premises, or generally false premises, are easily formalised in Aristotelian logic.

You formalize into a formal logic. Aristotle had only one formal logic, his syllogisms, and no syllogism has contradictory premises. So arguments with contradictory premises are not formalizable in Aristotelian logic.

This is your reasoning and it is based on two false assumptions.

Show me important theorems that could demonstrably be shown to be derived from mathematical logic and couldn't be derived by mathematicians using their natural logical capacity.

I give you a simple ECQ example:

It is true that I'm right;
It is false that I'm right;
Therefore, the Theorem of Bertini is valid.

Me, I can't infer that the Theorem of Bertini is valid from the premises, even though they are very simple and I am sure I understand them.
EB

 

[Speakpigeon]

Ypu have to specify what area of logic you refer to and define it with context before debating.

A philosophical definition.

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

Logic (from the Ancient Greek: λογική, romanized: logikḗ[1]) is the systematic study of the form of valid inference, and the most general laws of truth.[2] A valid inference is one where there is a specific relation of logical support between the assumptions of the inference and its conclusion.

That's the definition of logic as a field of study, not as a human capacity.

If the field is mathematical logic, then it's mathematics, not logic. Not in my book.

Starting with Aristotle and expending somewhat on his work, logicians, essentially the Stoics and the Scholastics, studied a formal method of logic.

I call "logic", i.e. logic proper, the natural capacity of the brain to make deductive inferences, as evidenced by the performance of most human beings.

In math, science, and engineering logic is an axiomatic system with postulates, axioms, and rules.

Axiomatic systems are axiomatic systems, not logic...

Unless you could prove, empirically, that the "postulates, axioms, and rules" are correct of logic as a human capacity.

Mathematical logic is a subfield of mathematics exploring the applications of formal logic to mathematics.

Prove to me that the expression "formal logic" is appropriate here. Formal system, sure, logic? I don't think so.

Are you refuting math over the last 200 years?

Show me important theorems that could demonstrably be shown to be derived from mathematical logic and couldn't be derived by mathematicians using their natural logical capacity.
EB

 

[fromderinside]

Wow. Man's natural logical capacity. Can you point out any competent psychologist that promotes such theory?

You do understand that human capacities are hierarchical topped of with first survive.

Surely you hand wave here.

 

[steve_bank]

That's the definition of logic as a field of study, not as a human capacity.

If the field is mathematical logic, then it's mathematics, not logic. Not in my book.

Starting with Aristotle and expending somewhat on his work, logicians, essentially the Stoics and the Scholastics, studied a formal method of logic.

I call "logic", i.e. logic proper, the natural capacity of the brain to make deductive inferences, as evidenced by the performance of most human beings.

In math, science, and engineering logic is an axiomatic system with postulates, axioms, and rules.

Axiomatic systems are axiomatic systems, not logic...

Unless you could prove, empirically, that the "postulates, axioms, and rules" are correct of logic as a human capacity.

Mathematical logic is a subfield of mathematics exploring the applications of formal logic to mathematics.

Prove to me that the expression "formal logic" is appropriate here. Formal system, sure, logic? I don't think so.

Are you refuting math over the last 200 years?

Show me important theorems that could demonstrably be shown to be derived from mathematical logic and couldn't be derived by mathematicians using their natural logical capacity.
EB

I see, before Aristotle there was no logic. People just made random choices. Everything builds on what came before. Aristotle did not create in a vacuum.


You talk about logic yet are unable to dwescribe the difference between induction and deduction. I had that in a logic class in philosophy.

You do not seem to have any understanding beyond syllogisms and formal logic which is actually quite trivial compared to everything that goes in the the realm of logic and reasoning. For example fuzzy logic I linked to.

You can not do a proof from scratch yet somehow you think math proofs and reasoning are all wrong.

You toss out net quotes yet you have yet to nmake a consistent reasined gogent stamen of problem and supporting logic.

Lets see some of that reasoning and logic you talk so much about.

Make an ironclad Aristotolain laogical argument impossible for us to disagree. That after all is the purpose of logic and argument is it not? Truth.

Use your logic to prove the truth of what you think. If you were philosophy in college you must have wrote papers.

Anything humans do is a fubction of the brain. Anything humans do is a human capacity. Logic in the end is a function of our brains. Unlees you are a pre 20th century philosopher who views mind as a metaphysical abstraction not tied to biology.

 

[fromderinside]

You can include 20th century neuroscientists to an exclusion to that notion entirely. The nervous system is a thing. The brain is a thing. Neural processing of self-other information takes place. The whole jumble can be forced in to some sort of model. But be assured, it does not do what we claim as we claim it does.

The brain is not an airplane. It is not designed. It is evolved so it actually doesn't have functions. It evolves processes which contribute to and add to our survival capability. Scientists and lay persons knowing only what we can do organize them to some purpose as it seems to be our need for communication. As far as I know for certain nervous tissue organizes by recruitment, inhibition, conduction perhaps develops associations through chemical and neural processes, and perhaps can by repeating similar activity within the nervous system as that sensed from the activity of others can learn and empathize, etc.

I'm intrigued by Crick and other's analysis of the evolving brain, the requirement for certain neural capacities be present for self-other representation and interaction. Structures within the brain show evidence they developed sequentially providing increased utility as they evolved primarily developing increased capability upward although there is mounting evidence for evolution of descending neural evolution as well.

 

[Speakpigeon]

I see, before Aristotle there was no logic. People just made random choices. Everything builds on what came before. Aristotle did not create in a vacuum.

You talk about logic yet are unable to dwescribe the difference between induction and deduction. I had that in a logic class in philosophy.

You do not seem to have any understanding beyond syllogisms and formal logic which is actually quite trivial compared to everything that goes in the the realm of logic and reasoning. For example fuzzy logic I linked to.

You can not do a proof from scratch yet somehow you think math proofs and reasoning are all wrong.

You toss out net quotes yet you have yet to nmake a consistent reasined gogent stamen of problem and supporting logic.

Lets see some of that reasoning and logic you talk so much about.

Make an ironclad Aristotolain laogical argument impossible for us to disagree. That after all is the purpose of logic and argument is it not? Truth.

Use your logic to prove the truth of what you think. If you were philosophy in college you must have wrote papers.

Anything humans do is a fubction of the brain. Anything humans do is a human capacity. Logic in the end is a function of our brains. Unlees you are a pre 20th century philosopher who views mind as a metaphysical abstraction not tied to biology.

Mindless derail.
EB

 

[fromderinside]

Derail? Probably. Mindless only to those who little capacity or credibility to judge.

Above should get you silly-gism machine working.

 

[steve_bank]

I see, before Aristotle there was no logic. People just made random choices. Everything builds on what came before. Aristotle did not create in a vacuum.

You talk about logic yet are unable to dwescribe the difference between induction and deduction. I had that in a logic class in philosophy.

You do not seem to have any understanding beyond syllogisms and formal logic which is actually quite trivial compared to everything that goes in the the realm of logic and reasoning. For example fuzzy logic I linked to.

You can not do a proof from scratch yet somehow you think math proofs and reasoning are all wrong.

You toss out net quotes yet you have yet to nmake a consistent reasined gogent stamen of problem and supporting logic.

Lets see some of that reasoning and logic you talk so much about.

Make an ironclad Aristotolain laogical argument impossible for us to disagree. That after all is the purpose of logic and argument is it not? Truth.

Use your logic to prove the truth of what you think. If you were philosophy in college you must have wrote papers.

Anything humans do is a fubction of the brain. Anything humans do is a human capacity. Logic in the end is a function of our brains. Unlees you are a pre 20th century philosopher who views mind as a metaphysical abstraction not tied to biology.

Mindless derail.
EB

Ok, lets get back to the topic. Clarify what you mean by 'wrong logic' when it comes to math.. Does it mean math is not based on solid logical foundations? Does it mean the wrong kind of logic?

Please show expplicut examples to clarify your response.

All's I have seen after all your posts is 'mathematicians have it wrong I have it right'.

Show a specific example of where a specific math theory or proof is wrong and what the right approach is.

Considering how long math has been used for real world problems across all of it and has not shown any problems where is your problem?

Keep in mimed all math truths are not derivable with Aristotelian logic. These days computer iteration techniques are used to search for a proof.

Give us a syllogism. P1 P2 therefore math is...

Hard to take you seriously when you do not know the difference between induction and deception.

 

[Speakpigeon]

Ok, lets get back to the topic. Clarify what you mean by 'wrong logic' when it comes to math.. Does it mean math is not based on solid logical foundations? Does it mean the wrong kind of logic?

Please show expplicut examples to clarify your response.

All's I have seen after all your posts is 'mathematicians have it wrong I have it right'.

Show a specific example of where a specific math theory or proof is wrong and what the right approach is.

Considering how long math has been used for real world problems across all of it and has not shown any problems where is your problem?

Keep in mimed all math truths are not derivable with Aristotelian logic. These days computer iteration techniques are used to search for a proof.

Give us a syllogism. P1 P2 therefore math is...

Hard to take you seriously when you do not know the difference between induction and deception.

Mindless drivel again.

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.

You don't understand the question, Steve. You really don't. In fact, you just don't understand English to begin with. Would? Would? Would?

And how do you expect that your response could be relevant when you don't understand the question?
EB

 

[steve_bank]

Ok, you say you do not know how bad it is. Can you give one example of how bad logic can affect math? Humor me, I am a little slow on the uptake.

I

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.

You seem to have a problem with mathematicians. Clarify the underlined.

Saying again, intuition and experience are used to construct the proof, once the proof is established it is validated by applying logic. A guess may be a starting point in the process followed by back and forth deception and induction.

It is not just math it is problem solving in general. There are no rules on how to apply logic to solve a problem.

You are stuck on syllogisms and simple logic. Read a book on how to do proofs.

In Calculus there are methods of integration, not all clear rule based logic. In integration by parts there is no way to no a priori if the technique will work for a specific equation. It requires trial and error and a staring point based on experience and intuition.

If it all was a matter of applying Aristotelian logic there would be no unsolved problems.

 

[steve_bank]

The word to describe a lot of math and general problem solving is heuristic. Not all math is derived by a mechanical logical process or algorithm what we call a 'plug and chug' process. The process of discovery and invention in science and engineering is heuristic. One starts with a guess and through trial and error arrives at a solution. The solution itself is not a guess or opinion. Once established a solution is subjected to logical scrutiny and validation by empirical test..

The question as to wither or not all math proofs can be devised by and an algorithmic logical process goes back at least to the early 19th century.

The final proof of any math is empirical testing.

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

A heuristic technique (/hjʊəˈrɪstɪk/; Ancient Greek: εὑρίσκω, "find" or "discover"), often called simply a heuristic, is any approach to problem solving or self-discovery that employs a practical method, not guaranteed to be optimal, perfect, logical, or rational, but instead sufficient for reaching an immediate goal. Where finding an optimal solution is impossible or impractical, heuristic methods can be used to speed up the process of finding a satisfactory solution. Heuristics can be mental shortcuts that ease the cognitive load of making a decision.[1]:94 Examples that employ heuristics include using a rule of thumb, an educated guess, an intuitive judgment, a guesstimate, profiling, or common sense.

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


In calculus, and more generally in mathematical analysis, integration by parts or partial integration is a process that finds the integral of a product of functions in terms of the integral of their derivative and antiderivative. It is frequently used to transform the antiderivative of a product of functions into an antiderivative for which a solution can be more easily found. The rule can be readily derived by integrating the product rule of differentiation.

If u = u(x) and du = u′(x) dx, while v = v(x) and dv = v′(x) dx, then integration by parts states that:


Strategy[edit]

Integration by parts is a heuristic rather than a purely mechanical process for solving integrals; given a single function to integrate, the typical strategy is to carefully separate this single function into a product of two functions u(x)v(x) such that the residual integral from the integration by parts formula is easier to evaluate than the single function. The following form is useful in illustrating the best strategy to take:

 

[Speakpigeon]

Reminder.

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

 

[fromderinside]

Neuroscientists dispute the notion there is something called human intuition. Such runs counter to scientific findings of the means by which humans came about. You need to operationalize your declaration of human intuition. Nothing you say makes sense unless you at least perform that rather fundamental task.

 

[steve_bank]

To me intuition is a distilation of knowledge and experience. One comes to solutions of problems without going through a technical logical process.

A non technical manger came to me with a problem. A bit later I came to him with the solution. He asked me how I derived the solution. I had no answer because I did not go through a logical analysis to begin with. He got pissed thinking In was playing with him.

I go to the point where I could go to the computer and start deigning circuits without any conscious thought, I had done so much of it.

I believe our brains wires learning and experience such that it is like an an log continuous computer as opposed to a sequential digital computer. Our neural net.

It is how past skilled craftsmen learned to do their work, iMO.

 

[fromderinside]

Weak on operations gee. Whether thought is conscious or no it depends on input and association (process) to produce output (product).

Some of us have strengthened our ability to reach back and find training that lead to designing such as circuits. I'm sure you could trace back through your training and practice and develop a fairly concise statement about how your product came to be. Your manager may begin to snore, but, you can do it.*

PS: Seems to me that one apparent objective the nervous system is continuously evolving toward is to make accumulated experience produce action that is less and less concentrated effort to do that. That is reduce experience to producing something near reflex. If such were not the case then the faster reacting auditory system would not be the sense category driving the head to turn the face toward sudden nearby sounds. Nor would there necessarily be peripheral visual mechanisms and the ability to concentrate on them to sudden movements in darkness.

*a^{_{fter all concise is a relative term}}

 

[Speakpigeon]

Intuition
1. The faculty of knowing or understanding something without reasoning or proof.

People demonstrate the reality of intuition by their objective performance in everyday activities. Having a conversation is a complex activity. You can't think about what you want to say and how to say it at the same time that you would think about the grammar of what you say and whether the sense n which you use 99% of the words you use is appropriate. Ergo, intuition: The faculty of knowing or understanding something without reasoning or proof.

We understand each other, and to achieve this, we have somehow to infer the meaning of what people say as well as of what they don't say. You don't perform this very complex activity through any conscious reasoning. Ergo, you use your intuition. Intuition is a fact of our mental life.

Most of the intuitions we have relate to what we have trained ourselves to understand. Intuition in this case is only possible if we have trained our brain into the domain concerned. However, the case of logic is different. You don't have to train yourself. Logic is entirely intuitive.
EB