korona wrote:What is the physical importance of the existence of arbitrary products in the category of sets? That's an axiom we use constantly but its only justification is that our theorems become more beautiful. Sometimes we just choose our axioms so that we get more general theorems.
It looks to me like you are arguing that Set Theory, in which the only objects are sets, is independent of physical references? Perhaps you are also arguing that ZFC is rigorously defined, has no physical references and can be used to construct most other axiomatic systems? The lack of foundation that you perceive is that there is no absolute reason to choose ZFC over some alternate systems? But having chosen ZFC, everything that follows is unambiguous?
I have no problem with any of the above. My argument is not that there is no way to prefer one set of axioms over another.
My argument is that it is not possible to unambiguously specify any set of axioms in the first place. [Edit: I've been lumping "meaning" in with the definition of axioms. It is
possible to define a set of rules (symbol manipulations) unambiguously. It isn't
possible to assign meaning to those rules unambiguously. This is a fairly important distinction that I now realise I haven't been making clear. Sorry for this mistake. I can now see that mathematicians tend to put weight onto the symbol manipulation over specific meaning for given symbols. As such - at least some of the following is arguing against a strawman.]
ZFC cannot be defined unambiguously. The words and symbols used to describe the axioms of ZFC are not defined within mathematics. Their meaning derives (possibly indirectly) from our experience in the physical world. And that experience is subjective, cannot be communicated directly, is partial, and in no way provides an unambiguous platform on which to build unambiguous systems.
As far as I can tell this isn't the "foundational crises" in mathematics.
When defining terms ("meaning") we do so by reference to other terms. But those other terms are defined by reference to yet further terms.
If any set of terms can be constructed in an unambiguous manner then we can construct further terms from those initial ones (axiomatic mathematics).
But if those initial terms are in any way ambiguous, then anything we construct from them will also be ambiguous (cross-referencing may help to reduce ambiguity...).
If we have no starting point (trans-dimensional alien), there is nothing to build upon. No way to create communication ("meaning").
We do have a starting point... us and the physical universe we inhabit. But we don't have a full and complete understanding of that starting point. Any "meaning" that we derive is incomplete, partial, ambiguous. Anything we build on that is also ambiguous.
Without a fixed, definite starting point there is no way to create a fixed, definite set of axioms. We cannot create ZFC in an unambiguous fashion.
That some of the systems that have been created work is a defense, to a degree. But to pretend that ZFC is a definite, unambiguously defined system is going to far. It isn't a question of finding inconsistencies within ZFC... it is that ZFC cannot be properly defined in the first place.
Hilary Putnam has certainly addressed this point... so it is absolutely known to mathematics (e.g.):Twin Earth Thought Experiment
From the referenced wiki page:
In practice, most mathematicians either do not work from axiomatic systems, or if they do, do not doubt the consistency of ZFC, generally their preferred axiomatic system.
So, you are certainly correct that this isn't news to mathematics. And "it works, bitch" is a fair summary of the approach mathematics takes.
But at the same, time, anything based on axiomatic mathematics (no matter how indirectly) is nothing more than a statement of our biases. And no doubt there is value in understanding what our biases are....
However, I had the idea that there was a mathematics that existed independent of humans. That there was a realm of concepts not tied to our physical existence that could be explored. It turns out that modern mathematics doesn't even attempt to address anything that isn't intimately tied to our biases. I find this disappointing.
So, I'm going to make my own mathematics (with blackjack and hookers) which isn't hobbled by the impossibility of defining a fixed point upon which to build everything else.
MartianInvader wrote:But your objection to mathematics applies just as equally here. What does "predicts the universe" mean? How can we know what a prediction actually is? How can we know it actually came true, or did not come true? How can we know what the values of measurements really were? At some point, we have to just buckle down and say, "Look, we can all agree that we understand what we mean when we talk about 10 grams, 1 centimeter, 3 seconds, etc., and we'll base our studies on this mutual understanding and go from there." So it goes with math, and pretty much any other discipline for that matter.
I agree that physics is no better founded than mathematics. My point was simply that physics doesn't claim to be well founded, whereas I was under the illusion that mathematics had loftier ideals. I even thought that mathematics might strive to reveal "universal truths". As it turns out, mathematics is just about finding better ways to count money.
There's nothing wrong with that, per se. It is my illusions that are at fault for regarding mathematics as something it isn't.
And your last point is extremely relevant. It isn't obvious that there is an alternative. This is a hard limitation on how knowledge can be expressed and communicated. Lacking any alternative, we work with what we have no matter how flawed. Doing nothing is not an option.
However, I think it is possible to construct something that doesn't depend on a fixed starting point that axiomatic mathematics requires.
Treatid wrote:We must attach "meaning" or "intention" to a bit string in order to work with it (e.g. "equals", "greater than", "contains", "addition").
I reject this assertion. We can work with a set of symbols and rules for manipulating them without understanding them. Computers do it all the time. Computers are commonly used to generate and check proofs, but I don't think anyone will claim that the computer understands the meaning of the symbols it's manipulating.
We constructed the computer (and the program it runs). That construction is where we assigned meaning.
The computer is just manipulating symbols. You are right - the computer does not perceive any meaning in those symbols. It simply does what it does according to its structure.
But by the same token, a computer does not check proofs, we may interpret the output symbols as a proof.
The symbols never hold any meaning themselves. The symbols are just labels that we attach to meaning. A computer manipulates symbols. It does not manipulate meaning.
Meaning is abstract. We have no way of directly expressing or communicating meaning. We can communicate symbols.
Now it's true that we do attach meaning to symbols when we're working on them; this is what makes math useful. But the definition of an axiomatic system is independent of and not reliant upon this meaning.
Ah... we could express a set of axioms as a computer program. The hardware and software are known deterministic systems. By expressing the axioms within this deterministic system, we are unambiguously specifying a set of axioms for a system. There is no argument over what the system will do (run it and see). There is no sense in which this system is ambiguous.
Except for my last sentence, I agree.
But the "attaching meaning to symbols to make math useful" bit is a fairly essential part of the process. And it is this part that cannot be done unambiguously.
It's also true that we choose which axioms to use based on the meaning we assign to them and (generally) our real-world intuition. So what?
So - we cannot communicate the meaning we have attached to a symbol. If the meaning you attach to a symbol is not identical to the meaning that I attach to a symbol then the meanings we each arrive at after manipulating that symbol are also likely to differ. Yet we have no way of detecting that difference - unless the different meanings lead to different choices of manipulation.
You are right: So what if my "meaning" for infinity is different to your "meaning" for infinity. If we are both constrained to the same set of manipulations then it doesn't matter if our meanings differ... we should both still output the same sets of symbols.
Hmm... your argument is good. It does help me to see why mathematicians are content to largely ignore the "meaning" with respect to a set of axioms - and why "it works" is regarded as a sufficient argument.
But I don't think it contradicts my argument. It still leaves mathematics as an arbitrary reflection of our biases with no means to reveal anything that isn't a direct result of those biases. And since mathematics does understand that axiomatic mathematics doesn't have a solid foundation...
Which still leaves me wanting more than "an arbitrary set of manipulations" with no inherent significance beyond that which we place there.
Cleverbeans wrote:This is not true. Non-Euclidean geometry being an easy first example.
I think the example you give shows that we do not always understand the rules/meanings we assume from the real world - and that careful examination can reveal our misapprehensions. I would say that the discovery of non-euclidean geometry is very much rooted in our real world experience.
The rest of what your post I agree with and is a good summation of both what I (naively) wanted to say and some of my position.