Theoretical physicist Paul Davies wrote:
But what are these ultimate laws and where do they come from? Such questions are often dismissed as being pointless or even unscientific. As the cosmologist Sean Carroll has written, “There is a chain of explanations concerning things that happen in the universe, which ultimately reaches to the fundamental laws of nature and stops… at the end of the day the laws are what they are… And that’s okay. I’m happy to take the universe just as we find it.”
Assuming that Davies is correct, I find it odd that there is little interest for understanding the laws of nature. There are some interesting questions to be answered, such as: Where do the laws come from? How do they cause things to happen?
Physicist Neil Turok once posed the question:
What is it that makes the electrons continue to follow the laws?
Indeed, what power compels physical objects to follow the laws of nature?
The question I would like to focus on is: what would a naturalistic explanation of the laws of nature look like?
Frankly, I don’t know where to start. What I do know is that a bottom-up explanation runs into a serious problem. A bottom-up explanation, from the level of say bosons, should be expected to give rise to innumerable different ever-changing laws. Different circumstances, different laws.
But this is not what we find. Again, Paul Davies:
Physical processes, however violent or complex, are thought to have absolutely no effect on the laws. There is thus a curious asymmetry: physical processes depend on laws but the laws do not depend on physical processes. Although this statement cannot be proved, it is widely accepted.
If laws do not depend on physical processes, then it follows that laws cannot be explained by physical processes. IOWs there is no bottom-up explanation for the laws of nature.
But what does it mean for naturalism if there is no bottom-up (naturalistic) explanation for the laws of nature? How does the central claim ‘everything is physical’ make sense if there is no physical explanation for the laws of nature? What if it is shown that the laws of nature control the physical but are not reducible to it?
You mean to tell me that you can invoke the law of non contradiction on a mathematical definition without laying it out in propositional form, to infer a contradictory conclusion? That’s what a syllogism is in my book
…this somehow reminds me of what I read here
“Inferential relations are always between items with propositional form.”
Inference has nothing to do with it. Definitions must be logical. Or you could say definitions can be illogical and that’s okay because you don’t care. In which case my obligations to explain things to you cease to have effect.
Finding a logical contradiction in a definition is not a formal inference?
Precisely because I care, I’m asking YOU to show what that contradiction is.
You still haven’t answered if you affirm the laws of logic. This tells me how much you really care.
Erik’s mistake stems entirely from his misconstrual of the word “collection”.
To Erik, a set is a collection, and a collection isn’t a collection if there’s nothing in it.
Informally, that is true. If I have no stamps, I won’t offer to show you my stamp collection. (If I had only one stamp, I wouldn’t call it a “collection” either, yet Erik oddly seems to have no problem with sets containing a single member.) But mathematicians are smarter than Erik. They understand how to formalize and generalize concepts in a useful way. Expanding the concept of “set” to include the empty set is extremely useful, just as expanding the concept of “number” to include zero was extremely useful.
Under the expanded concept, there is nothing oxymoronic about the empty set. In effect, Erik is simply whining that mathematicians have borrowed a word from the vernacular, modifying and formalizing its definition to suit their purposes. But thinkers do this all the time, across a wide range of disciplines. Erik’s complaint is bogus.
Not sure what you mean by “affirm”. If they are self-evident truths? If that’s what you mean, the answer is I don’t know (again, Sellars seems to suggest there’s no such thing, as opposed to everyone else before him, but I’m not in a position to form an opinion), but I don’t think it matters at all. You made the claim that there is a logical contradiction, and we are both working under the assumption that the laws of logic are in place when we are talking about logical contradictions. Whether that assumption is provisional or foundational is entirely besides the point. One can work with different axiomatic systems at different times even if they are not consistent with one another. The universe won’t collapse because of that, believe me
“In mathematics, a set is a well-defined collection of distinct objects, considered as an object in its own right.” https://en.wikipedia.org/wiki/Set_%28mathematics%29
So, there are objects in a set. The question arises: What object is there in an empty set? This is unanswerable, which is why they made it an axiom, as if there were no contradiction to be explained. This is one philosophically suspect aspect of it.
Another suspect aspect: Why construe numbers from sets? What is there in numbers so that they need to be construed? And why from sets? Is this an analogy (in which case numbers and sets are not of the same species, they bear only incidental resemblance and it’s unjustified to say that numbers are generated from sets) or a true construal (in which case there must be a demonstrable special relationship between sets and numbers, so that one can generate the other)? Nobody has even attempted the latter, except by assertion or postulation of axiom, so it could just as well be the former. I’m fine with the former.
Hunh.
So you don’t know whether to affirm the law of non-contradiction. If this is an honest answer, then we are done. If this was not an honest answer, you have a chance to change your mind and then I must test if you really mean what you say.
What do you mean by affirm?
It means: Understand what it means and abide by it.
whatever
I’m still trying to wrap my head around the idea that someone thinks that there can’t be sets without members.
When I say, “there are no giraffes in Kansas”, what I’m saying is that the intersection of the set of all giraffes and the set of all things in Kansas has no members. And that’s extensionally equivalent to the empty set. It’s the same thing as saying “the number of giraffes in Kansas = 0”.
The idea of a set without any members is just a translation, into an extensional semantics, of the idea that we can conceive of things that don’t exist.
So denying that it makes sense to talk about sets without members is basically saying that either (1) it doesn’t make sense to say that we can conceive of things that don’t exist, or (2) it doesn’t make sense to put into an extensional semantics the idea that we can conceive of things that don’t exist.
Erik,
You’re making my point for me.
Without realizing it, you are making the following argument:
But why on earth should mathematicians be limited to using your definitions? They’re smarter than that. They generalize the notions of “set” and “collection” to include those containing no objects, just as they generalized the notion of “number” to include zero.
You hit a problem and came to a screeching halt. Mathematicians solved the problem and kept going.
You’re no mathematician, Erik.
Perhaps you are taking that too literally. That’s not really a definition. Rather, it’s a motivating description.
Any mathematician would want to count EVEN, the even integers, as a set. But nobody can go about collecting them. Similarly, any mathematician would want to count ODD, the odd integers, as a set. And again, nobody can go about actually collecting them. And if EVEN and ODD are sets, then their intersection has to be a set. And that intersection would be the empty set.
Note the wikipedia quote is about a “well defined” collection. There are huge philosophical disagreements about “well defined”. That’s part of the disagreement between Intuitionistic mathematics and traditional mathematics.
No. To say that the box is empty is equivalent to that there are no things in it, but to say that there are ghosts or unicorns in it is saying a whole different thing.
I’m getting the impression that your confusion is genuine and profound. It wasn’t just a slip at first. You hate metaphysics with passion.
My definition? How about the Wikipedia definition? Does it define a set as a collection of objects or doesn’t it?
Now you lost my trust completely. Either you have a source or you are making things up as you go.
Why not?
That’s what foundations of mathematics does. It allows set theory to be a foundation for everything.
Personally, I am not a foundationalist. I don’t believe that mathematics requires a foundation. But I still see foundations as a useful exercise. If nothing else, it demonstrates the versatility of set theory.
Erik,
As I just said, you are assuming that a collection must contain at least one object:
Mathematicians are smarter than that.
Yeah, to avoid an incoherence they make an axiom out of it. When Christians do the same thing with the trinity (a concept that apparently says 1=3) do you call it smart? By the current standard, you do.
So you are not really asking the question. Nor answering it.
Erik,
No, they merely extend the definition of “set” to include those containing no members. It’s an obvious generalization, it’s extremely useful, and it leads to no incoherence whatsoever.
They solved the problem. You got stuck.
Erik,
Again, keep in mind that mathematicians are smarter than you. The fact that you’re baffled by these questions is just an indication that you aren’t a mathematical thinker.
Mathematicians like axiomatic systems, and ever since Euclid they have appreciated the value of minimizing the number of axioms and keeping them simple.
Defining all of mathematics in terms of set theory is one attempt at achieving a minimal, elegant set of axioms. Category theory is another candidate.
Kantian Naturalist,
It’s just plain silly, isn’t it? Especially when he insists that there’s a logical contradiction in the definition.
But I think the question about the law of non-contradiction is a much more interesting one, and since you are here (LOL), may I ask how would Sellars deal with Erik’s dogmatic take on the laws of classical logic? I mean, we all live our lives by them, (for the most part?), but one can also work on the basis of paraconsistent logic.
What does it mean to work under the assumption of certain axioms (classical logic, paraconsistent logic) in terms of their epistemic status? Would those systems be in our “space of reasons” simultaneously despite of being inconsistent with one another?
…well, I’ll leave it there. Been rewriting this for over an hour and I’m still not sure if those questions make any sense, so feel free to ignore them
ETA: well, I guess classical logic and paraconsistent logic would only be inconsistent with one another if we assume the law of non-contradiction… ugh
I’m just puzzled that you are getting your panties in a bunch over something so unimportant.
For me, “set” is defined by the axioms. Most commonly, those are ZF (Zermelo Fraenkel). But the axioms don’t define what a set is. Rather, they define what operations you can apply to a set, and what rules are followed in applying those operations.
A motivating description gives an example of something that fits the axioms, and suggests what the operations amount to for that particular example.
If you don’t see the power of set theory to clarify our thinking about concepts, that’s not really my problem.
I should say, however, that I am not an extensionalist about concepts, and indeed I think that extensional semantics was a profound error of 20th-century analytic philosophy. But it was, in a ‘cunning of reason’ sort of way, an extremely important error. Without it, we would be lacking a robust understanding of the difference between formal and natural languages.
I know why you think this, but you are (as usual) completely mistaken about my views. I love metaphysics, and I take it very seriously. My favorite metaphysician is Gilles Deleuze, with Dewey being a close second and Spinoza a distant third. I’m a scientific metaphysician (by way of methodology) and a process ontologist (by way of ‘substance’).
I simply don’t think that pre-Enlightenment rationalism, which is your gig, is a viable project. In the wake of Kant, Hegel, Peirce, Darwin, Nietzsche, Frege, and Wittgenstein, pre-Enlightenment rationalism is the philosophical equivalent of sticking one’s fingers in one’s ears and saying “I can’t hear you!”
I see the power along with its limits.
Similarly, set theory can be overhyped, probably better suited for something other than generating numbers. I genuinely appreciate some mathematics, such as geometry, topology, and chaos theory, so I have my points of comparison.
Your own attitude towards pre-Enlightenment rationalism seems to be consisting in a lot of “I can’t hear you!”. Yours is the easy option, because we live in a different era where pre-Englightenment rationalism is long since past and forgotten. Whereas for those who adopt pre-Enlightenment rationalism as their system of choice, it’s a very conscious choice after having given some alternatives a go. In this era nobody is born into pre-Enlightenment rationalism, but rather matures into it. So you should have a better explanation for why you think pre-Enlightenment rationalism is not viable.
No, YOURS is! In other words, I know you (both) are, but what am I? I mean, I have my fingers up my nostrils, and I can’t hear either of you!
🙂
dazz,
These are very good questions! To my knowledge, no one working in the Sellarsian tradition has written about non-classical logics, and logicians who do work in non-classical logics don’t engage with Sellars. But I will ask around.
My conjecture is that Sellars could happily be a pluralist about logics, classical and non-classical alike. Each can function as an independent language-game. Sellars occasionally refers to what he calls “the polydimensionality of discourse”. Our discourse about the world involves different ‘dimensions’: the mathematical dimension, the logical dimension, the moral dimension, the aesthetic dimension, the empirical dimension, and so forth. (Sellars, being a committed atheist, never mentions a religious dimension, but there clearly is one.) Each dimension has its own constitutive rules that determine what is a permissible assertion within that dimension. For example, within the logical and mathematical dimensions, an assertion is permissible only if it is deductively provable. In the empirical dimension the constitutive rules are quite different.
The difference between classical and non-classical logics lies in the inferential moves available from a contradiction. In classical logic, anything follows from a contradiction (P & ~P –> R). Different non-classical logics have different rules on how to handle contradictions, but they still have rules about avoiding incompatible commitments. I think that a Sellarsian could say that non-classical logics add a new language-game to the logical dimension of discourse.
What characterizes each dimension as being part of the space of reasons isn’t our commitment to “the laws of thought”, but rather to our commitment to keeping track of what others (and ourselves) are committed to saying and entitled to say. (This is a thought of Brandom’s that he develops in Making It Explicit and subsequent tomes.)
In that sense, the space of reasons is certainly governed by norms, but they are pragmatic norms rather than what is codified by Aristotle. The fundamental notion here is “thou shalt avoid incompatible commitments!”, but it’s a ‘thou shalt’ that lies in how we hold each other accountable for what we say and do.
Sellars is quite explicit that to be a human being is to live a life that is structured by norms. (Our lives are ‘fraught with ought’, as he puts it.) The harder question, then, is to explain the origins of norms in naturalistic terms. In “Philosophy and the Scientific Image of Man”, Sellars observes that the seeming impossibility of accounting for norms in non-normative terms is what keeps creationism alive.
I think that our counter-parts at Uncommon Descent would agree. Quite a few of them are convinced that rationality and morality — the two defining aspects of normativity — cannot be accounted for in naturalistic terms, and that naturalism must throw normativity under the bus. (The reason why Alex Rosenberg is their favorite naturalist is that he is perfectly willing to throw normativity under the bus!)
Erik,
You aren’t even comprehending the problem that mathematicians were trying to solve.
They weren’t looking for a convenient and intuitive way to generate numbers. They were looking for something that could serve as a foundation for all of mathematics.
Likewise, particle physicists don’t offer the Standard Model (or competing successor theories) as a convenient way of studying structural fatigue in bridges. They’re looking for a theory that can unify physics.
If set theory is to serve as a foundation for mathematics, it had better be able to generate numbers; and if a physical theory aims to unify physics, it had better be able (in principle) to explain structural fatigue in bridges. But it’s not about convenience or intuitiveness.
Kantian Naturalist,
Great stuff, thanks again KN
Neil doesn’t see it your way, and he’s a mathematician. Maybe I don’t comprehend the problem that mathematicians were trying to solve, but you are not the guy to tell me about it.
Erik,
Please educate yourself:
Pay particular attention to this paragraph:
And for good measure, take a look at this:
Yup, as I suspected – the aim of foundationalist mathematicians was to align themselves with the axioms, never questioning whether one or some of them might be arbitrary or unsound or perhaps stretched beyond their domain of application. Like Roman Catholics. I had no doubt this was the problem all along.
Erik,
You keep forgetting that the mathematical community is not a sea of Eriks bumbling about in a Dunning-Kruger fog. They’re smarter than you; they make progress while you remain stuck.
Reread our exchange regarding the empty set:
Erik:
keiths:
To me, this seems laughably absurd. I doubt that any foundationalist mathematician would agree.
That’s not even wrong.
Let alone philosopher of mathematics, epistemologist, or anyone who had actually studied any of this stuff seriously.
The quote keiths provided says what it says. Sort it out with him.
The quote I provided doesn’t say what you want it to say, Erik.
You are in full crackpot territory now, claiming that the entire mathematical community is deluded about the empty set, while you alone have seen it for the incoherent sham it is.
In reality, you’re just too dumb to figure out that they’ve generalized the concept of set to include sets with no members, just as they’ve generalized the concept of number to include zero.
You’re no mathematician, Erik.
Neil confirmed that empty set is as (un)problematic as square circle. He also knows what a definitional self-contradiction is. He didn’t deny this is the case. Instead, he claimed I am unacquainted with how mathematicians think. Actually, I know how mathematicians think, which is why I brought these things up and he has no way of denying them. By the same measure, he has no way of denying the trinity, its coherence or otherwise. All that is needed is to say to him that he is unacquainted with how theologians think.
You don’t know what generalization is.
That was never the plan in the first place.
Erik,
The only supposed problem you’ve identified with the empty set is that it conflicts with your personally preferred notion of “set”, in which there is at least one element in every set. But who cares? That’s not important at all.
What matters is whether the mathematicians’ notion of “set” works and is coherent. It does, and it is. It also avoids the awkwardness of your approach.
The mathematical community did it right. You screwed up.
Mathematicians’ notion says “In mathematics, a set is a well-defined collection of distinct objects, considered as an object in its own right.” I haven’t disputed this definition. Further, set theorists (not mathematicians in general) posit empty set as an axiomatic starting point. Nobody relevant here ever denied that this was the case.
You have your own idiosyncratic view of the situation.
The mathematicians’ notion of “set” encompasses the empty set. Yours doesn’t.
That makes sense. They’re better at this stuff, so they can see the advantage of the generalization.
You reject the empty set because you aren’t smart enough to apply the generalization. They generalize and are able to retain the empty set, making set theory much cleaner.
False. “Empty set” is a distinct notion from “set”. Neil has only said mathematicians have no difficulty with empty set. Well, neither do I. I have no difficulty with square circles either. I see nonsense for what it is, without difficulty. (And the trinity doesn’t make sense to me either. Subdivisions do.)
Again, it’s not generalization. It’s extension. You are supposed to be the smart guy, so it’s not good when I have to teach you.
keiths:
Erik:
To mathematicians, the empty set is a set. Denying that confirms your crackpot status.
Their inclusive notion of “set” works better than yours and is perfectly coherent.
You are stymied by a problem that mathematicians solved long ago.
Define set as per mathematicians.
For now, the definition quoted by me is the only one on the board, undisputed. If you want to sound relevant, stay up to speed.
Maybe this, from Russell’s Principles of Mathematics (1903)–and which he largely took from Peano–will be helpful.
keiths:
Erik:
Christ, Erik. Are you really that helpless?
From Basic Set Theory:
Not at all. You were failing to define what you were talking about, up to now. And you are still failing to see the contradiction. Neil never substantively disputed my point, only hand-waved it away because he is not a foundationalist. You are failing to see that too. And you are failing to see the further point with KN where it all started with. Too many fails in one.