# Thread: How a wrong logic could affect mathematics?

1. ## How a wrong logic could affect mathematics?

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

2. Can I ask what your background in mathematics and logic is?

3. Back in the early 20th century at a conference rhetorically Hilbert asked if all true mathematical propositions can be proven. The issue at the time was whether or not there was an inconsistency in the foundations of math lurking.

I believe in the 90s a systematic review from the bottom up was done.

In the end logic is consistent which means valid. Conclusion is followed from premise. Actual proof of logic is test in reality.

All math that has real world usage is verified by usage. No inconsistencies in any applied math has surfaced as yet. There is no way to say a problem will never arise. Same as any physical science theory, the proof is in the results.

Ann abstract proof of any logic is simply showing that using rules of logic the conclusion follows from the premise.

Try reading a short book How To Read And Do Mathematical Proofs. Read it in the 80s. It will answer all your questions by doing the exercises. You will never understand it as a philosophical abstraction.

4. Originally Posted by Torin

I'm not discussing mathematics in general (only mathematical logic).

I'm not discussing whether mathematical theories are logically consistent (I'm sure they are).

I'm also confident all methods of logic proposed in mathematical logic are logically consistent.

What I dispute is that any method of logic proposed in mathematical logic could possibly correctly model the logic of human beings. My claim is based on the existence of the paradox of material implication, which is at the foundation of mathematical logic since 1850. The paradox of material implication has been discovered by mathematicians themselves and it is in all mathematical logic textbooks.

Here is an example:

All men are immortal; therefore all squares are circles.

This is considered a valid implication in mathematical logic just because it is false that all men are immortal. If you believe that it is not true that all men are immortal, you should accept that the conclusion is valid. Me, I don't think that's how the human mind works. Nearly everybody would accept that this is just stupid, except possibly many mathematicians but mathematicians are biased precisely because of their training in mathematical logic: All mathematicians are human beings and all human beings are very probably biased by their education, therefore...

Another example:

Jo is a giraffe and Jo is not a giraffe; therefore Jo is an elephant.

This is again considered as valid simply because the premises are contradictory and therefore necessarily false. See? It is valid to infer that Jo is an elephant...

I have a few other reasons to dismiss mathematical logic as a joke but this one is plenty good enough.

To answer your question: my background in mathematics is minimal. I did two years in mathematics and physics at university and that was a long time ago.

Originally Posted by Torin
and logic is?
You ask about logic, not mathematical logic. My background in logic is that I am a human being. I don't see any good reason to assume that I would be less qualified to speak about logic than mathematicians. Further, my claims are quite minimal and it is rather obvious that they are correct. If you think not, please explain to me your thinking. I have been researching actively this subject for several months now and I haven't been able to find anyone who could provide a convincing counter-argument.

My background in logic is also Aristotle. You know, the guy who defined what people came to call "logic" for 2,500 yeras?

So, me, I read Aristotle and I see that his notion of syllogism properly defines my own intuitive notion of what logical validity is. You should start from there.

Concerning mathematical logic, I had a class in mathematical logic when I was a university math student. It was my first encounter with formal logic and I'm quite sure I never thought about logic as such previously. Looking at what the young teacher was telling us about formal logic, I just took it in initially without any problem. The conjunction, the disjunction, the negation. I was discovering the principle of defining these things through truth tables. No problem at all. In fact, all pretty damn obvious. Then came the turn of the material implication. I remember vividly: I looked at the truth table of the material implication and my brain just puked. I had no background in formal logic. There's no way I could have been biased in any way. And my reaction was entirely intuitive. I haven't changed since. My intuitive reaction is still the same. I also found several times in books written by professional logicians the observation that may students couldn't understand the material implication. I myself didn't say anything at the time. It is likely that most students who have a problem don't say anything so that the problem is under-reported.

You can also test yourself. Look at any truth table of various conjunctions and disjunctions. You should be able to notice your own intuitive reaction: all good. Now look at the truth table of various implications that are deemed valid in mathematics. Now, try to understand how the result is obtained. Well, me, I don't understand. It just doesn't make any sense. You know, formal logic is supposed to be a model of our intuitive sense of logic. How could mathematical logic be possibly correct and yet contradict our sense of logic?!

I understand for example the following argument: A square is a circle and all circles are round; therefore all squares are round. It's clearly valid even though we usually understand the premises and the conclusion to be false. So, I have not problem with validity in case of false premises. But I certainly have a problem with the idea that an argument is valid because its premises are false. This is plain idiotic.

And yet mathematicians have been peddling this idiotic notion of validity since broadly Boole, who sort of invented the material implication, so broadly since 1850. So, it's now nearly 170 years that (most) mathematicians ground their work on logic on this idiotic definition. 170 years, that's maybe 7 or 8 generations of mathematicians. There is in the world probably more than a million intellectual workers alive today. They would have been all trained on this basis! Mathematicians doing theoretical work on mathematical logic are much fewer but that's still probably a lot of people (plus computer scientists and philosophers). And yet, all they can do is material implication?! Whoa. I'm underwhelmed.

Please also note mathematical logic is a branch of mathematics, not a method. Mathematical logic includes all sorts of different methods of logic, "classical", paraconsistent, multivalued, relevant etc. Alternatives to mainstream, so-called "classical", logic, are even more pathetic so I won't discuss them here. There are too many of them to even know how many there are exactly. These methods adopt axioms and calculus principles that are often contradictory to each other so that they can't possibly be all correct. So, in effect, it's not just me saying mathematical logic is junk. It's many mathematicians themselves who prove my point.

Please note that mathematicians have produced different, and contradictory, theories of geometry, i.e. curved and flat. These theories are all logically consistent but they still contradict each other. We all accept that these theories are just that, theories. Whether they may be correct is an empirical matter and should be decided on empirical grounds, not on the say-so and authority of mathematicians who in reality have no more expertise on this than any idiot alive today. Same for logic.
EB

5. Originally Posted by steve_bank
Try reading a short book How To Read And Do Mathematical Proofs. Read it in the 80s. It will answer all your questions by doing the exercises. You will never understand it as a philosophical abstraction.
EB

6. my background in mathematics is minimal. I did two years in mathematics and physics at university and that was a long time ago...My background in logic is that I am a human being. I don't see any good reason to assume that I would be less qualified to speak about logic than mathematicians.
Ok then.

7. Originally Posted by Speakpigeon
Originally Posted by steve_bank
Try reading a short book How To Read And Do Mathematical Proofs. Read it in the 80s. It will answer all your questions by doing the exercises. You will never understand it as a philosophical abstraction.
EB
Same response as dozens of times in the past when you ask the same question.

All logic is the same matemetical or otherwise. If, and, or...

Symoloc systems vary. Boolean logic symbols in electronics or mathematical symbolic logic. It is all the same.

Logic can be valid in the by the rules conclusion follows from pro[positions. True of an elecytronic lojic or verbal syslogusm.

The 'proof' of anything is how it is reflected in reality There is a saying in spgtware, the logic in code alwys works perfectly, it performs as written. The problem is when logic does not do what the cider thinj]ks it does. Formal logic can never be invalid or wrong when rules are properly applied.

The application of logic can be wrong only when it is improperly applied to a problem.

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

The only proof is in testing. Can there possibly be same problem in the rules if logic not yet mahifested, there is no way to say. The Incompleteness Theorem may apply.

You really, really do not grasp logic. And you really do not seem to wasn't an actual answer.

8. Originally Posted by steve_bank
The 'proof' of anything is how it is reflected in reality (...)

Formal logic can never be invalid or wrong when rules are properly applied.

Originally Posted by steve_bank
The only proof is in testing. Can there possibly be same problem in the rules if logic not yet mahifested, there is no way to say.
So, you say here there could be a problem with logic, apparently, but then you just said formal logic can never be wrong. Sorry, but no one will understand you. Could you rephrase or explain yourself?

Originally Posted by steve_bank
The Incompleteness Theorem may apply.
I'd love to see you explain this one.

Originally Posted by steve_bank
You really, really do not grasp logic.
LOL.

Originally Posted by steve_bank
And you really do not seem to wasn't an actual answer.
Sorry, I can't parse that.

Originally Posted by steve_bank
Good, so explain to me what's wrong with it.
EB

9. There is no way to say conclusively there is no problem lurking in the foundations. If you worked real world problems instead of simple syllogisms you might understand that.

As a logician are you familiar with the Incompleteness Theorem and its implications? It would be pointless to engage you on this. It has real world imp,ications in algorithms.

I'd say your understanding of logic is like understanding the syntax and grammar of a language but being unable to understand how to read and comprehend. You say mathematicians have it all wrong. Can you go from the geberal to a specific example where there is something wrong? Or are you just trying to eleveate yoursdelf?

What does A AND A! mean, What dies A OR A! mean?

10. Originally Posted by steve_bank
There is no way to say conclusively there is no problem lurking in the foundations.
Yes? But it's you who said this:

- Formal logic can never be invalid or wrong when rules are properly applied.

- The application of logic can be wrong only when it is improperly applied to a problem.

You seem unable to say anything without contradicting yourself at every turn.

Originally Posted by steve_bank
If you worked real world problems instead of simple syllogisms you might understand that.
What do you know of me?! You think I spent all my life considering "simple syllogisms"?! Are you for real?!

Originally Posted by steve_bank
As a logician are you familiar with the Incompleteness Theorem and its implications? It would be pointless to engage you on this. It has real world imp,ications in algorithms.
You don't have to explain anything. We all know you can't even explain yourself.

Originally Posted by steve_bank
I'd say your understanding of logic is like understanding the syntax and grammar of a language but being unable to understand how to read and comprehend.
And how could you possibly know that?

Originally Posted by steve_bank
You say mathematicians have it all wrong.
No, I didn't. I'm talking only of mathematical logic and only to the extent that it is understood or presented as a correct model of the logic of human reasoning. Boole talked of the "Laws of thought". Frege thought he could formalise mathematical proof. They had it all wrong.

Originally Posted by steve_bank
Can you go from the geberal to a specific example where there is something wrong?
I already gave a specific example. It's a very well know example, not even my own. So what's wrong with you?

Originally Posted by steve_bank
Or are you just trying to eleveate yoursdelf?
You know, in most countries it's probably a crime to advocate the rape of children, don't you?

Originally Posted by steve_bank
What does A AND A! mean, What dies A OR A! mean?
There's no difficulty there. Even Boole could get that right. Now, please explain to us how the implication works... Nobody did, you know. So if you can do it, please show the world.
EB

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