Are formal logic and mathematics incompatible?
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andyandy
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in what way are they incompatible? Mathematical proof is built on formal logic. Indeed, formal logic is maths.
Mangafranga
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Isn't the domain of maths numbers, while the domain of logic is not limited to numbers?
Or at least isn't there an interpretation of "maths" such that it is the study of numbers? (And another which means something like "the study of something using mathematical techniques")
Or at least isn't there an interpretation of "maths" such that it is the study of numbers? (And another which means something like "the study of something using mathematical techniques")
Robin
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Here is the exchange from http://www.internationalskeptics.com/forums/showthread.php?postid=2788230#post2788230
Mobyseven said:
andyandy
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It seems strange to baulk at the use of logic with regards to rigorous mathematical operations yet acccept that same system with regards to far less rigourous linguistic operations. Predicate logic is simply an appropriation of mathematical logic.
[latex]$ \forall {x}, {x} \in h \rightarrow x \in m [/latex]
This is mathematical logic. That it also represents
All JREFer are immortal.
andyandy is a JREFer.
Therefore andyandy is immortal.
shouldn't suddenly require that we regard the two as far removed from each other.
Mangafranga
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Not that this important, but doesn't that only represent the first premise of the argument?
andyandy
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let's see....
it's only a tweaked version from here
with h = set of JREFers
m = set of immortals
a = andyandy
oh yes
[latex]$ \forall {x}, {x} \in h \rightarrow x \in m [/latex]
[latex]$ a \in h [/latex]
therefore
[latex]$ a\in m [/latex]
blobru
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Robin said:
My understanding is that formal logic comes from philosophy; it wasn't applied to math until Frege in the 19th century; out his work came mathematical logic, the study of valid forms of math proof.
Where you say "[t]he problem starts when you try to apply formal logic to anything except
Mobyseven said:
Formal logic and mathematics are not equivalent, but there is overlap; they are certainly not "incompatible". I take Mobyseven's point to be it is very difficult to describe all of mathematics in formal logic, which is true enough (see Russell and Whitehead's monstrous Principia Mathematica, which required over 1000 pages to prove 1+1=2). However, formal logic assumes that the entities it describes are well-defined, which is why it models proof in the 'domain of numbers' (the conclusion is true), but only models argument in the 'domain of words' (the conclusion is valid). Formal logic may be applied to either, but its conclusions are more powerful in math than in language.
...and boy, am I embarrassed.
My understanding of the situation was that for mathematical operations to be described using predicate logic functions had to be 'imported' into the system, thus causing the divide. My understanding was that the 'hierarchy' was as such:
Seems I was very wrong about that. I plan to visit the man who told me such when I am at uni on Monday and slap him.
Apologies, Robin, you were correct. Can anyone give me a basic rundown on how the mathematical notation is 'translated' into predicate logic notation?
Does that sentence evn make sense now?
My understanding of the situation was that for mathematical operations to be described using predicate logic functions had to be 'imported' into the system, thus causing the divide. My understanding was that the 'hierarchy' was as such:
Logic
|
|
/\
/ \
Predicate Logic Mathematics
|
|
/\
/ \
Predicate Logic Mathematics
Seems I was very wrong about that. I plan to visit the man who told me such when I am at uni on Monday and slap him.
Apologies, Robin, you were correct. Can anyone give me a basic rundown on how the mathematical notation is 'translated' into predicate logic notation?
Does that sentence evn make sense now?
Mangafranga
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I'll try and lessen your embarresment by giving an answer to your question (thus opening myself up to be wrong).
The question makes sense. I can give you half an answer. That is I can't tell you how they define numbers and such in logic (except that you should look for info on set theory), but I can give you a translation of the formula you gave in the other thread into predicate logic anyway. The problem is that most of the meaning is exported into the predicates (i.e. the D and the S). I understand this is not really an answer to your question, but I thought it might help you to understand some of what is going on with predicate logic. I have used (x) to mean "for all x", <-> to mean "iff", = means "identical with", and AND to mean "and". Also note that my iff translation of the "=" of maths is my own, that is it seems right to me, but I don't know if it's commonly used.
(x)Dx <-> (z)Sz AND x = z
(whereby D=dy/dx, and S=z^3 - z^2)
To actually get the numbers and such into the notation itself is what takes the few hundred pages of Principia Mathematica that blobru alluded to. This http://plato.stanford.edu/entries/set-theory/primer.html#4 might be a good place to start to give you and idea on how they define numbers.
Robin
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That is not an occasion for embarrassment, it is good critical thinking to take on board new information. It is something that I am often fail at.
So I am trying to do likewise and look up the history of logic in my Funk and Wagnalls.
Aristotle invented classical logic in 3rd century BC. The mathematical logic of definitions, axioms, theorems and proofs was developed by the Greeks from 6th to 5th century BC. This mathematical logic included many of the features that we associate with formal logic today such as modus ponens, proof by contradiction.
The two strands don't appear to meet up until 19th century, with people like Boole and DeMorgan.
So I guess the situation was not as simple as I was making out either.
So I am trying to do likewise and look up the history of logic in my Funk and Wagnalls.
Aristotle invented classical logic in 3rd century BC. The mathematical logic of definitions, axioms, theorems and proofs was developed by the Greeks from 6th to 5th century BC. This mathematical logic included many of the features that we associate with formal logic today such as modus ponens, proof by contradiction.
The two strands don't appear to meet up until 19th century, with people like Boole and DeMorgan.
So I guess the situation was not as simple as I was making out either.
Robin
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I wasn't quite clear on this at the time. I knew that Euclid used some constructions that are part of what we call formal logic, such as proof by contradiction. My reading since indicates that the definition/axiom/theorem/proof structure of mathematical logic was developed in the 6th and 5th century BC.
Yes I accept that correction.
Again, you put it better than me. Mind you there are better systems than PM, ZF for example.
blobru
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Could it be you're subconsciously jealous of all the attention the Stundie nominees are getting, so you decided to go for one yourself?
(That's one of the funniest threads ever, btw -- proves there really is nothing funnier than TRUTH!!!)
Robin: good catch about mathematical logic going back to Pythagoras and Euclid. Not "mathematical logic" in the modern sense, which is Frege extending the symbolic syntax of Boole and De Morgan to describe math, but they certainly invented techniques like reductio ad absurdum (...contradiction) to prove root(2) irrational, no largest prime, etal. (And axiomatic deduction as well, now that I think about it, which so impressed Socrates he tried to apply it to morals, which influenced Aristotle's thought and syllogistic logic; so in a roundabout way logic does come from math too).
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Godel Numbering may be just the tip of the iceberg. Wolfram (creator of the Mathematica software and author of A New Science, Kind Of) conjectures that all systems of elements (symbols, states, etc.) manipulated by rules, other than those that behave in obviously simple ways, are equivalent in their computational power. This strongly suggests that they can also all be transformed into one another, as Godel numbering transforms predicate logic into number theory.
Respectfully,
Myriad
Yiab
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If by "formal logic" you mean "predicate logic" (i.e. quantifier logic), then they are not incompatible. All of mathematics (as far as I know) is expressible within second order predicate logic.
If by "formal logic" you mean "boolean propositional logic" then the question boils down to an unanswerable one: is mathematics consistent?
Boolean logic is provably consistent, Gödel proved that we cannot prove the consistency of mathematics in a similar way, hence if mathematics is internally inconsistent then it is incompatible with logic.
If by "formal logic" you mean, by some bizarre twist of fate, you mean "constructionist logic" or "minimal logic", we formally get the same reduction as with boolean propositional logic above, but the question becomes more interesting as we do not necessarily know that constructionist or minimal logic are consistent.
If by "formal logic" you mean something extremely vague, then they are not incompatible since there are many different ways of expressing your vague notion in precise ways within mathematics.
Logic is a subject area within mathematics and even philosophical logic relies heavily on mathematical logic these days. Modern mathematics uses logic very heavily at its foundation and, in a sense, mathematics is simply an exploration of a variety of special cases of that logic, so they can easily be seen as one and the same.
If by "formal logic" you mean "boolean propositional logic" then the question boils down to an unanswerable one: is mathematics consistent?
Boolean logic is provably consistent, Gödel proved that we cannot prove the consistency of mathematics in a similar way, hence if mathematics is internally inconsistent then it is incompatible with logic.
If by "formal logic" you mean, by some bizarre twist of fate, you mean "constructionist logic" or "minimal logic", we formally get the same reduction as with boolean propositional logic above, but the question becomes more interesting as we do not necessarily know that constructionist or minimal logic are consistent.
If by "formal logic" you mean something extremely vague, then they are not incompatible since there are many different ways of expressing your vague notion in precise ways within mathematics.
Logic is a subject area within mathematics and even philosophical logic relies heavily on mathematical logic these days. Modern mathematics uses logic very heavily at its foundation and, in a sense, mathematics is simply an exploration of a variety of special cases of that logic, so they can easily be seen as one and the same.
cyborg
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Oh, I am quite familiar with this. My friend has a copy of the book which I have been dipping in and out of (currently reading GEB in its entirety first).
The idea of computational equivalence has been around since Turing's machines were shown equivalent to Church's Lambda calculus.
Absolutely. This is what interpreters are all about and I respectively submit that since any computation model you wish to name can, and almost certainly has, been programmed into a humble PC that evidence for this is pretty strong.
However the OP's question was specifically about whether or not formal logic and mathematics are compatible. Godel Numbering shows how a purely typographical system, such as predicate logic, can be converted into an algebraic system. As such the question, I feel, is pretty roundly answered.
Agreed. They're not only compatible with one another, they're equivalent to one another (since to prove the former, you prove the latter).
The interesting part of Wolfram's case is not the ideas or definitions of computational universality or computational equivalence, which as you say are long-established fundamentals of computing theory. It's the conjecture that all systems capable of exhibiting complex behavior, including analog systems like three-body problem cases and analog computers, and even including quantum systems, are computationally universal and computationally equivalent. This also implies that all types of mathematics will ultimately have similar severe limits in their ability to summarize such systems, which explains why so many simple-seeming math questions (such as the three-body problem, conjectures about prime numbers, Diophantine equations, etc.) turn out to be intractable.
In conventional computing theory, computational equivalence is something that is proven (usually by exhibiting a conversion from one system to another). Wolfram speculates that computational equivalence and universality apply to all systems capable of exhibiting complex behavior, even where we don't yet have any idea how to prove it.
Respectfully,
Myriad
Well, this discussion has certainly turned interesting very quickly, if a little bit over my head! Thanks Mangafranga for the translation and thanks Robin for pointing out I was wrong in the first place.
I'm hopefully going to transfer to a science degree next year, wanting to study maths, so eventually all this stuff should make sense to me! ...I hope.
I'm hopefully going to transfer to a science degree next year, wanting to study maths, so eventually all this stuff should make sense to me! ...I hope.