Question for discussion:
Is mathematics more fundamental than logic, or vice-versa? Neither? Or is it more complicated?
93 thoughts on “Mathematics and logic”
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The Skeptical Zone
"I beseech you, in the bowels of Christ, think it possible that you may be mistaken."
How can mere numbers or what numbers represent be elevated to the higher intellectual processes of logic and investigation??
Math is just a language of reality and not reality itself!
Its primitive(primate) ideas that a simple language of things is equal to the essence of things.
Logic and investigation leads to the truth of a complex universe.
People are not described by math nor is anything in the universe.
Its only a raw language of very basic boundaries.
In fact i say teaching kids math has interfered with the intellectual grpwth of same kids especially when trying to persuade them to a interest in nature and its study or science.
Math is a waste of intellectual effort unless discovering new things in math. very few and seldom.
More biology and less math would of seen evolutions death years ago.
A bit like looking for the origin of life. There appear to be memes buried so deeply in our cultural history that we assume them without thinking about their origins.
Maybe you should post this over at UD! 🙂
Oh….
Neil:
keiths:
Neil:
I’m using the standard definition, akin to the one Bruce quoted.
That’s not the right question. We can never know all of mathematics, so that is hardly a knock against set theory.
The real question is this: If you knew every knowable truth of set theory plus predicate logic, would you automatically know every knowable truth of mathematics?
That’s irrelevant, because the reductive project is mathematical, not historical.
Why is that a problem? Nothing about reduction requires us to use “main examples” or historically significant ones. Set theory can “build” the numbers from scratch.
Reciprocating Bill:
Did you grow up in the Big City? The New Math tsunami didn’t wash over my little Indiana hometown until about ten years later. I was one of the weird kids who actually liked it, but overall, it was a pedagogical disaster.
Parents hated it because they couldn’t help their kids with their math homework. And you can imagine what it was like for the poor teachers who were suddenly confronted with set theory after teaching the 3 R’s for 30 years.
If that stuff hurts your head, stay away from the axiom of choice and resulting theorems like Banach-Tarski Paradox
Is arguing about the meaning of words relevant?
If you want to address the issue that the SEP quote popularizes, I think you gotta do the math!
After all, this stuff isn’t philosophy!
That’s why I went on to mention the motivation behind the original logicist project. It’s more than just picking some axiomatization that works – that has been done already. The axioms are required to be “fundamental” in a sense that goes beyond the requirements of supporting known mathematical structures. Whether this is a reasonable idea, and whether it can be realized is a still-open question in the philosophy of mathematics.
Ah, but as a great Zen master once said, it is only by considering ideas and their contradictions that we come to a synthesis and the truth.
No, wait, is that Zen or Hegel? Sometimes I get them mixed up.
First time commenter. I have tried to follow the comments and it looks like you are discussing good things so I thought I would add my two cents. It’s probably already been covered 🙂
A computer is created out of electronic circuits. The parts of the computer that matter can be formed out of logic gates which duplicate boolean logic. You can create all of the required gates out of OR and NOT, as demonstrated by minecraft computers (http://www.joystiq.com/2010/09/30/working-16-bit-computer-built-inside-minecraft/). Such a computer is made out of redstone and at the most basic level that only provides OR (two inputs directly connected and provided as an output) and NOT (an input connected to a redstone torch which then provides an inverted output).
Programs within the computer are describable using discrete mathematics. All programs that can be expressed in a turing machine have equivalent expressions in discrete mathematics.
Programs within the computer are able to simulate and answer almost all mathematical problems (see https://en.wikipedia.org/wiki/Hypercomputation for some currently uncomputable problems).
So a computer built out of two boolean operators is able to perform almost all of mathematics. In this system that is produced you can simulate a working computer (see the minecraft example above).
Given this ability to model mathematics in terms of (boolean) logic, and the ability to model boolean logic in terms of discrete mathematics, I think that there is no more fundamental side of things. I do not see this as a chicken and egg thing, where one must exist before the other can, as each can be expressed in terms of the other.
I do realise that this conversation has covered predicate logic, which is something that I should read more about. I also realise that discrete mathematics is but a branch of mathematics as a whole. I am really just trying to say that you can express one in terms of the other (and vice versa).
From looking at his books on Amazon, I would guess Wright falls on the philosophical side, sad to say for his job prospects on Wall Street.
Seriously, I guess the serious point of my original reply classifying people as philosophers or mathematicians was this: if fields of knowledge are philosophy until they have enough rigor to be math (or enough empirical content to be science), then Frege and Russell are in the transition but still on the philosophy side for logic, whereas Godel is on the math side.
I do realize it is a continuum, not a sharp divide. But nothing to do with the continuum hypothesis, of course.
On the issue of writing style. I think the technical literature for philosophy is mostly natural language with a smattering of predicate or modal logic in formal notation. Math is various cryptic arrangements of Greek and other foreign alphabets and invented symbols. And science is math (including stats) plus diagrams plus natural language. So I think it would be hard to compare readability of the technical literature.
I’ve tried the Logicism entry at SEP but I have not got much out of it yet. But if I understand your post correctly, I think it gets at a core issue.
The standard axioms of set theory are enough to derive the rest of mathematics in a technical sense which is acceptable to the consensus of mathematicians who have examined those technical arguments.
Further, these axioms of set theory are accepted by mathematicians for pragmatic reasons: they lead to fruitful, elegant, useful mathematics.
On the other hand, Logicism is (partly) about finding deeper, philosophical reasons to use these axioms.
Reminds me of a recent thread on the nature of morality and moral values, but ’nuff said on that.
No, you clearly would not.
Those are part of what make mathematics an interesting subject.
Such a computer is able to do just about all of the computations. Whether that is the same as doing just about all of mathematics is what is being argued in this thread.
What is the difference between being able to do almost all the computations and being able to do almost all mathematics
Geometry (broadly conceived).
https://en.wikipedia.org/wiki/List_of_interactive_geometry_software
I don’t know what you mean by broadly conceived.
Neil is getting metaphysical.
Well my original point was that you can implement discrete mathematics in boolean logic. You can also implement boolean logic in discrete mathematics. I used computers as an example of doing exactly this.
I am no mathematical philosopher so there probably are problems with my idea. If so I would like to hear about them!
Hi orfax,
Welcome to TSZ.
Neil has a lot of odd and sometimes contradictory ideas on these topics, so if you find yourself scratching your head, you’re not alone. His answers also tend to be terse, so it can take a long time to figure out what he means.
Having interacted with him for some time now, I may be able to shed some light on what he’s trying to say in this thread.
Neil thinks that brains don’t primarily compute. Instead, he thinks that the major component of thought is categorization, which he sees as a non-computational process implemented via our sensory interactions with the world. He also seems to see categorization as a geometric process, perhaps because when you create a category, you divide the world into two pieces: inside the category and outside, which can be visualized geometrically.
With that background, you can begin to see what he’s getting at in this thread. If thinking is mostly categorization, and if categorization is a geometric process, then most of our thinking is essentially geometric. However, axiomatic geometry is of course computable. Since Neil wants our intelligence to be beyond the reach of mere computation, he needs parts of geometry to be non-computable.
Hence his reference to “Geometry (broadly conceived)” as being non-computable.
It doesn’t make a lot of sense to me, but that’s what he’s trying to say, as far as I can tell. I’m sure he’ll correct me if he thinks I got his viewpoint wrong.
Mathematics is about proving theorems from (interesting, useful, productive) axioms. (*)
I agree that Turing machines can do proofs. Also, at some point, an AI will be able to create theorems. (If you are worried about Godel’s Incompleteness theorems, see Franzen’s book for the refutation of the arguments from people like Lucas or Penrose).
So far, so good. I agree with your points.
But you need some axioms to start with to do mathematics. There are no axioms to start with in bare predicate logic. Or at least not a rich enough set to get you all of mathematics: calculus, topology, abstract algebra: to infinity and beyond if you like!
So the question is, what is the minimal set of axioms that make sense to mathematicians and which can also be used to derive the rest of mathematics (where “derive” is defined in a technical sense by the consensus of expert mathematicians working in that area).
The agreed answer by those experts is: the standard axioms of set theory.
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(*) ETA: Mathematics is not about doing computations in software to get some mathematical result. It is about doing proofs of theorems. But I do agree that proofs can be done by software and eventually some AI will be as good at that as human mathematicians
Topology is part of geometry (broadly conceived), though it is not part of Euclidean geometry. The calculus has a geometric component (slopes of curves, areas under curves, etc). All of analysis (which includes calculus) is generally considered to combine algebra and geometry.
It is not a matter of what I want. Rather, it is a matter of what best accounts for human cognition.
Logic is a formal theory of a solipsist’s world. Geometric thinking (broadly conceived) is what allows us to connect to reality.
If you know the axioms of set theory and how to do proofs, would you be able in principle to derive the rest of mathematics?
I understand that the experts say “yes you would”, assuming the technical definitions of “derive”, “in principle”, etc that they use.
It depends on what you mean by “derive”.
Point set topology is usually discussed in terms of sets. But a truth of topology is not theorem in set theory. The basic axioms and definitions of topology are not logical consequences of the axioms of set theory.
Set theory (with logic), together with an additional set of conventions (the definitions and axioms of topology) would be sufficient.
Good point and one I had not thought through. The popularization I quoted from SEP also included the point that the definitions of any other branch of math (and I assume the axioms which use them) could also be stated using the concepts of set theory (along with the axioms of set theory and predicate logic).
Yes, they can probably be stated using the concepts of set theory, but they cannot be logically derived from the axioms of set theory.
That’s about what I meant when I said that I see set theory as a kind of universal modeling clay.
A few points:
Propositional logic refers to using operators like AND, OR, NOT to connect statements. It is the Boolean logic you refer to.
Predicate logic adds the ability to assign properties, to have variables, to have named entities (basically constants), and to use ALL, and THERE EXISTS. As in: ALL Men are Mortal, Socrates is a Man, etc. Or for a mathematical example, if you have done introductory calculus then I am sure the phrase “For ALL epsilon greater than zero, THERE EXISTS delta such that….” will bring back warm memories.
So you cannot do math without predicate logic. But you can do predicate logic without math.
Hyper-computation is an interesting theoretical concept, but I think the consensus is that nothing physical could be built to implement the things it discusses.
In any event, I don’t think you need hypercomputation to implement AI. Plain computation done by Turing machines is enough. However, the way I define computation for AI purposes is not the way Neil does, based on our past exchanges. Rather, the definition is based on thinking which is not found in traditional AI (see here if you are interested).
Nonetheless, I believe that extended definition of computation is implementable to create AI by Turing machines. Of course, that is just my belief: it is still very much an open question being researched.
Maths got emergence?
It depends on what you are looking for.
There are many dualities in mathematics. One example would be the Fourier transform, which maps something from the physical domain (such as a sequence of vibrations) to something in the frequency domain (a combination of frequencies (or tones or wavelengths). The transform is reversible.
When we use a Fourier transform, what looks like a reduction on one of those domains is transformed into what looks like emergence in the other domain.
Neil,
Of course the axioms of topology aren’t derivable from the axioms of set theory. If they were, we wouldn’t consider them to be axioms!
Also, mathematics includes subfields with differing and sometimes contradictory axioms. For example, Euclidean geometry depends on the parallel postulate, and non-Euclidean geometry depends on its denial. If the axioms had to be derivable from the axioms of set theory, then set theory would have to be inconsistent in order to accommodate both Euclidean and non-Euclidean geometry!
I was about to ask if there are choices in mathematics. The obvious answer is no, but I wonder if that is an artifact the way the universe selects and shapes our perceptions.
Thanks for this explanation.
I guess this is where I went wrong. I viewed mathematics as the set of consistent systems built on the axioms as described previously. I was trying to demonstrate the equivalence of two different systems (propositional logic and discrete maths) by showing that they can be implemented in terms of each other. The idea of future research and discovery in these spaces was not considered.
As an aside, I do believe that software could be written for existing computers that would form an AI. My mention of hypercomputation was more to say that there are limits to what can be computed, and those limits may break the equivalence I am trying to show.
Anyway thanks for all the responses, really interesting stuff!
Your point about “implementable in terms of each other” raises a deeper issue, I think.
If computers are just Boolean machines, how can they “do” math? Is that the same as saying math is just Boolean algebra? Now, we don’t really know that computers can do math in the full way that humans can, it is still an open question,so maybe we don’t really need to answer that question yet.
But people can do math. And people are “just” properly organized and embodied electrochemical reactions. So is math just electrochemical reactions?
It’s the mind/body or syntax/semantics problem in some sense. Is the math just electrochemistry, properly organized? Is the mind just the body? How to get semantics from syntax? That’s been rather a popular topic at TSZ. And elsewhere.
I have thought about this before, and what computers require is time. The boolean operations that the underlying hardware perform step through different states. The discrete mathematics the programs represent also transition through states.
In my (limited) experience of mathematics this is unusual. A lot of the equations and theories I see are essentially invariant with time – they either hold or they do not.
I meant the word “do” in a different way, as I tried to explain in the rest of my post.
Be careful about intermixing physics equations with math. I’d associate ideas about time with physics, not math on its own.
I think you need to replace “reactions” with “processes” there.
Right.
I need to have more caffeine before typing these things I guess.
What if you include neither a parallel postulate nor its denial? Or when we talk about the the denial of the postulate are we just saying that it’s not the case that lines must remain parallel? Thx.
walto,
Then you have what’s known as “absolute geometry”. The downside is that you give up much of the richness (and many of the practical applications) of both Euclidean and non-Euclidean geometry. For example, the angles of a triangle are no longer guaranteed to sum to 180 degrees if you jettison the parallel postulate.
It’s a bit more complicated. In spherical and elliptic geometry, parallel lines don’t exist at all; that is, two arbitrary lines will always intersect. In hyperbolic geometry, given a line and a point that isn’t on the line, there will always be more than one “parallel” line through the point. In Euclidean geometry, of course, there is exactly one such parallel line.
Thanks.
Mathematics is the language of science. It is not logic. It can convey logic.