Let's equate logic and math

Philosophy Forums Philosophy
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I’m listening to a lecture series on the philosophy of language by John Searle from Berkeley University (I seriously recommend them–Searle is incredibly entertaining and funny–anyone who finds philosophy dry won’t after listening to his free iTunes-downloadable pod casts). He talks about Friedrich Frege, the father of modern language philosophy, and he says that the philosophy of language grew out of Frege’s attempt to equate mathematics with logic.

Now this thread is not about the philosophy of language, so I won’t go into the details of how the philosophy of language grew out of this, but I will briefly explain (assuming I understand Searle (and assuming Searle understands Frege)) how Frege thought mathematics relates to logic, and also why it failed spectacularly.

It starts with what Frege thought was a “scandle”–no one knows what a number is–“Quick–” Searle says, “gun to your head–what’s the number 2?” So Frege tried to define it. The number 1 is the set of all sets containing 1 member, the number 2 is the set of all sets containing 2 members, the number 3 is the set of all sets containing 3 members. To put it more simply, the number 1 is all singular things, the number 2 is all couples, the number 3 is all triplettes, etc. This brings mathematics into the sphere of set theory, and set theory is an outgrowth of logic–voila!

Why it failed? Russle’s paradox. If you can have sets of sets, you can talk about the set of all sets that don’t contain themselves. Does that set contain itself? I’ll leave it to you to wrap your head around that question. It also failed because of Godel’s incompleteness theorem. I’ll admit that I don’t understand why Godel’s incompleteness theorem disproves Frege’s definition of numbers (on which his equating math with logic rests), but Searle says so, and whatever Searle says is right.

Anyway, now to the point of the thread: why is this holy grail of mathematicians and logicians so ellusive? Why can’t we equate mathematics with logic? My own thoughts on the matter lead me to the following (which I don’t consider a full equating of math and logic, but I do think it’s a reasonable start):

Logic is a just a set of rules for thinking properly–modus ponens, Demorgan’s law, disquotation, the law of excluded middle–all these logical rules are just the rules that thought develops for itself as it perfects its function of getting its analysis of facts and propositions and arguments right.

But so are the rules of mathematics. With math, however, it’s a set of rules for how to think properly about numbers. That is to say, it’s the same activity of thought–figuring out the rules for getting things right–only applied to something much more specific: quantities.

But why does mathematics seem like a much more vast and elaborated system than logic? That’s a question I have no good answer for. I would propose that maybe it might have something to do with the happenstance stultifying influence that the middle ages had on Arostotelian (and basically all of classic intellectual) thought. Aristotle’s logic, which is very basic by our modern standards, was thought to be a done deal for the longest time (even a philosopher as recent as Kant said so) and the dominant methodology for studying the classics at the time (i.e. scholasticism) really avoided innovation (I don’t know if they actually forbidded it–but I know their religious views at the time did). Meanwhile–literally during those same middle ages–mathematics made some incredible strides, particularly in the department of algebra.

Why? I don’t know. Maybe it was because they couldn’t trace mathematics to any one particular sage or historical figure (Euclids maybe, but I believe he was famous for intruducing geometry, not so much mathematics, but I could be wrong). Maybe this made it seem like they weren’t blaspheming against anyone for changing or adding to the system. Maybe it was because mathematics is just so invaluable for so many applications in a way that logic isn’t–it does wonders for accounting, for war, for construction and engineering, for science, etc.–and so development and innovation was that much more needed in mathematics than in logic. But why is this? I’ve always wondered why logic, even with its modern mathematical-style notation (which Frege invented, BTW), doesn’t seem nearly applicable to anything as mathematics seems. Maybe we can touch on this somewhere in the course of this thread (if it goes anywhere at all ;P).

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Tragicly logic isn’t math based, but can indeed in some cases made math based. It requires enourmous effort to make a computer do a simple conversaion and even the most powerful computers fails at it.
In principle programmers should be able to make conversations a math based logical matter, but to actually conversate it takes enourmous intellect and rationallity to comprehend suble meanings of words which programmers has a hard time making computers understand, we often change a meanig with a word by changeing the tune.

Once the comprehension barrier is dealt with, we must be very careful as specially androids will be far better at conversating than us, and pray that we don’t get demagogues androids as many naive people will uncritically follow it.

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No, but the attempt is to make math logic based–that’s already done with computers.

My point is that if you make a list of all the rules thought stumbles upon to think properly, then you can take a subset of those rules (which happens to be huge) and see that it only applies to numbers. This seems too simple a solution to be what Frege, Russle, Wittgenstein, and company were all attempting to do. Am I missing something?

The question for me at this point is: what is it about numbers that makes them special enough to warrant so many specialized rules? Is it just that we’ve developed mathematics so much more throughout history than we have logic?

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gib, Numbers seem more concrete to me than language. 1+1=2 no matter what your language is. Multiplication and division (functions [?] of addition and subtraction will always be consistent, no matter what. Language isn’t like that. Language is fluid and protean.

Logic, imm, is a rhetorical ‘method’ that depends on “understanding by majority vote.” I’m stretching my point, obviously. People use language to communicate what’s basically a ‘feeling’ or belief. Logic arranges that language in a reasoned construct that really has nothing to do with “truth.” It has to do with validity based on language use and reasoning–along with bias.

Please keep in mind that these are my thoughts, only.

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Yes, I suspect this is why mathematics seems to have developped a lot more than logic. Numbers are concrete because you can point to instances of quantities in the concrete world–4 cows, 5 cars, half a pizza, etc. Math is a lot more demonstrable than propositions such as “Ignorance is bliss,” etc.

Really? That’s sounds Quinian. I would think logic is what thought attains to after it has perfected its methods of getting things right–and this involves testing beliefs and conclusions against reality. My grade 10 math teacher gave us an example of the logical fallacy they call “affirming the consequent”: All serial killers were breast fed as babies. I was breast fed as a baby. Therefore, I’m a serial killer. Doesn’t seem to make sense, does it? That’s why they call it a fallacy. But this fallacy–affirming the consequent–is a very easy one to get tripped up by–we all do it without knowing it, and probably more often than we think. We come to erroneous conclusions based on faulty logic, but we are corrected when we come face-to-face with the actual state of things in reality. We come across someone who we know is not a serial killer yet was breast fed as a baby, and we react by rethinking our line of reasoning. After several encounters like this, the mind learns the mistakes it makes in its thinking and refines the rules by which it guides thinking. This is how logic evolves AFAIC.

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Ideally, I agree. I don’t believe, however, that that’s how most people use logic (if they use it at all) in thinking. I think most people stop with, “This is what I think” or “I feel this is true.”

To go off on a slight tangent, using your example; (I wouldn’t have done this as a 9th grader, but…) what if I’d been there? I’d ask, “How do you know all serial killers were breast-fed? What if one or two had been adopted, or cared for in a foster home, where the mother wasn’t able to breast feed?”

Imm, logic is dependent on the first proposition in the syllogism, the truthfulness of which is considered either true or not true. That’s what I meant when I said it depends on a majority vote. I’ve called logic a rhetorical method because logic is a basis for formal discussion of a given subject. This is what a debate is. Using the presidential debates as an example, I’d guess most of the broad audience has already decided how they’re going to vote and their vote really depends on what they feel is ‘true.’

But to get back on track, hopefully. Logic as a rhetorical method is formal logic; logic used otherwise is informal logic. The most influential outcome, imm, of informal logic is critical thinking.

Critical thinking uses everyday language to examine, question, evaluate, and come to conclusions about things presented in everyday language. That makes it difficult for most people to practice because of the differing meanings and individual nuances of our language and because–and this is most important–to be successful in using critical thinking for self-reflection, a person has to be honest with her/himself.

As for maths and logic, isn’t symbolic logic a way of putting logic into mathematical format? Could AI be a way of mathematizing informal logic? On reflection, that may be why, if it’s true, I haven’t really accepted the idea of AI.

Btw, which Quin were you referencing when you said I sounded Quinian?

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You pretty much can treat symbolic logic like math. Once you get into language, problems being, but one can fuss around with the symbols like one can with mathematical figures.

But what is it you are hoping for/missing? What is the yearning here?

There are a lot of maths, doing a lot of different things, many very self-contained and not clearly related to other parts of reality, for example. And many of these maths need to get really complicated and or mathematicians are exploring, without any sense of application - though applications and coherence with other stuff may arise later.

The ‘math’ of logic, is fairly simple. I mean most of the stuff we use in arguments. I mean it has nothing on calculas or number theory or trig or whatever. So it would be odd if logic were as vast and complicated as math is. There are some other logics, but really the main one one uses in discussions is pretty simple. Once your throw in language, discerning the logical steps or lack of them can get very tricky, but that’s another cannister of parasites.

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No, it isn’t how most people use logic. But when I talk about “logic,” I’m talking about formal logic (the kind you’ll find in university level text books), not folk logic.

My theory is that what logicians are talking about when they talk about “logic” is the rules that thought would follow if it were perfect at getting things right. But since nobody’s perfect, no one really thinks in formal logic, at least not all the time. There’s a lot of talk about “my logic” or “your logic” or “fuzzy logic” or “monkey logic” which insinuates there can be different kinds of logic, but they’re all “informal” logic as far as I’m concerned.

Note that logic isn’t about the truth or falsehood of the propositions. So the following argument…

…is a perfectly valid logical argument. The second premise is false, but that’s neither here nor there. That’s why logicians like to deal with the mathematic notation you mentioned–it frees them from any bias over the truth or falsehood of the propositions.

And this may be yet another reason why logic hasn’t made as many strides as mathematics. It’s all fine and dandy to have a mathematics-like notation system and a set of rules with which to manipulate it, but if people can’t agree on the propositions–their thruth or falsehood, their meaning, how to phrase them–then you can’t even get started. Or if you do get started, but you don’t like the conclusions you come to, there’s a tendency to want to fudge the logic somewhat.

Yes, and this was Frege’s invention. But even with the notation, logic is about propositions, math about numbers. Frege’s project was an attempt to show that the latter was a more narrow field of the former.

You mean making a computer think with folk logic? I suppose, but I don’t think that’s what AI engineers are aiming for. I don’t think it’s their aspiration to get computers thinking with “faulty” logic, although it may be an inevitable side-effect if they ever bestow computers with the typical human mental characteristics that usually muddle up our otherwise logical thinking (things like emotions, desires, intuition, creativity, etc.).

I was thinking about this in light of this thread, and I realized I had the answer right in front of me all along. The gap between logic and mathematics had been closed since the invention of computers. Computer circuitry is based on logic (literally–computers are nothing put wires connected with “logic gates” as they call them), and built on top of that are math processors (adders, multipliers, counters, etc.). But I still don’t think the kind of logic you get with these processors is the kind of logic we go through in our heads when we mentally carry out math operations, so I still think my theory holds.

en.wikipedia.org/wiki/Willard_Van_Orman_Quine

Willard V.O. Quine was an American philosopher who we could call a “logic-skeptic”. If I understand him correctly, he would say that all the basic rules of logic aren’t written in the stars or even in our brains. They are rules that we adopt and even decide upon given whatever works for us. If our environment turned out different, we may have needed to come up with an entirely different logic in order to get by (affirming the consequent may have been a logic rule rather than a fallacy).

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Due to the uniqueness of individuals, each will come into enlightenment in his own way. You cannot turn out enlightenment on an assembly line because nature uses no model in creating living organisms. Rational thought does have its place though and one aspect of it is predictability. It makes it difficult to live in the world intelligently without a sense of expectedness and logicality. But not all give priority to certainty or anything associated with it simply because there can be none when everyone is looking for it.

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Mathematics is merely a subset of logic dealing with quantities.
…no magic or mystery to it.

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Let’s equate logic and math

[size=130]“Been there, done that, Wrote the Book, Didn’t really work”[/size]

                         Bertrand Russell, on [i]Principia Mathematica[/i]
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That’s exactly what I said. Glad to see I’m not the only one who sees this as obvious.

Did they really talk like that back then?

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Actually Hobbes influenced both Frege and Russell.

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Really? How so?

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Through his ‘calculus’ of words. Hobbes was the first to suggest computational linguistics, but he never got very far with it.

You might find more about it if you look up the history of linguistics. It may have been something I read in re-familiarizing myself with Frege. :slight_smile:

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That Bertie was a real modern guy uh?

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Math in itself is only what i call “liniar” logic, when neural networking attempt abstract logic, dunno if that answers anything of your questions.

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Anything dealing with equalities is logic. Mathematics deals with precise quantities and thus gives a greater variety of manipulations and methods to manipulate than merely stating simple equalities as is most common in what we call “logic”. They are the exact same thing except that in math, each term being considered can be arranged into other terms and portions with already known properties such that the logic can be immediately applied without having to specify the properties of the portions.

All numbers have certain properties in common. Mathematics merely takes advantage of that fact so as to make rules that circumvent having to specify logically why one can multiply (for example) both sides of an equation by the same number without violating the equality. Math is a shortcut to the actual logic that would have to be applied due to formalized rules for operators that have already been logically canonized.

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 And so was Leibiitz, and who came first is debatable.  But the relationship between a rartional and a quantitative logic may not be derivetive, but coincidentally surprising.
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I dunno either. I don’t really understand it. I would think all logic is abstract, not just mathematics. And what is “non-linear” logic?

That sounds pretty insightful, James, and I sense a hint of truth to it (I think you’re on to something), but you might have to unpack that a bit in order for me to fully appreciate it.

You mean Leibniz? Leibniz came first by about 200 years.