http://mashable.com/2016/06/02/elon-musk-simulated-reality/#sdLXHm2_jsqB
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The Skeptical Zone
"I beseech you, in the bowels of Christ, think it possible that you may be mistaken."
BruceS,
I really don’t know what you’re saying there, Bruce. What would the extension of a formula be if not its truth-value? And I added the ‘necessary,’ because as I read that definition, its not true for contingent formulae, as explained in my post.
Again, none of that seems to me to have anything to do with Quine. I generally agree with keiths’ remarks here. The whole discussion of ‘intensional’ and ‘extensional’ (and Quine’s positions on those matters) here has been (as keiths posts have helped explain) passing strange.
I ask again, what would it mean for physics not to be expressed using math?
I can’t tell if you’re making a psychological claim there or are agreeing with Kant that 5+7=12 is a synthetic apriori proposition.
walto,
I’m not commenting on Kant, or Quine, or psychology, or synthetic a priority.
I’m simply saying that mathematical formulae, except for the most trivial (like “7=7”), rely on intension.
There is intension even in your simple example, because “5+7” is an intension — that is, it specifies a property of the number it is referring to rather than naming the number outright. The right-hand side of the equation, by contrast, does name a number explicitly — “12” — so it is extensional. (Although a pedant might argue that even it is intensional, since “12” is just shorthand for the expression “1 * 10 + 2 * 1”. Please, let’s not go there.)
Here’s how I put it to KN earlier in the thread:
In your example, “5+7” is just shorthand for “the number having the property ‘is the sum of 5 and 7’.” It’s an intension.
Perhaps. I’m saying math is extensional in the sense of avoiding intensional contexts regardless of the fact that math terms can be defined intentionsally. And that type of extensionality is what Quine wants.
I agree that intensional definitions are used for math terms as in the formula you provide. But all that is saying is that we can use intensions to get at the extensions. For math to lack intentional contexts, all we need is to be able to substitute terms with same extension without changing truth. How we “get to” that extension, what intension (or Fregean sense) is being used to do that, does not matter.
Consider the standard English example I gave in my earlier post. “Sam Clemens” and “Mark Twain” have the same extension, even though their meanings differ. But when we substitute them into sentences involving psychological attitudes, the truth value can change. Hence those contexts are not extensional.
I’m saying you cannot have such intensional contexts using just math formulae.
Now I wanted to say one more thing.: If you allow English, then you can introduce an intensional context. For example, if Joe thinks sqrt(49) is 8, then
Joe believes 6+1 = 7
is true but
Joe believes sqrt(49) = 7
is false even though the extensions of 6+1 and sqrt(49) are the same. I mention that situation as this is how I think of your point of mathematicians employing conjectures.
In your last line you say “it’s an intension”–but I’m not sure what the “it” is referring to. Earlier you said “there is intension in [my 5+7] example” and that the 5+7 IS an intension. But, of course, if 5+7 and 10+2 ARE themselves intensions, then 5+7 isn’t really equal to 10+2.
I note too that you say that “12” is a name and is thus extensional. But, supposing you don’t take intensions to be the same thing as extensions, 5+7 will, on your view not be equal to 12 as one is an intension and the other an extension. I think these types of considerations should show that none of the the numbers can themselves BE intensions if any mathematical identities are to remain true.
To be fair, however, you also make the weaker remark that formulae RELY on intensions. That could be a psychological remark, as I said. It could mean that people need intensionality of some kind to do math. That seems right to me, but isn’t responsive to anything Bruce or Neil have said, I don’t think. (I have no idea what KN was saying.)
Finally, you say “5+7” is just shorthand for “the number having the property ‘is the sum of 5 and 7’.” But consider. One doesn’t need intensions to quantify into that context. The x which is such that it has the property of being equal to any y that is the sum of 5+7 is also equal to any y that is the sum of 6 + 6. There’s no problems with intensions there (as there would be if we stuck “the number of trees on Lombard” in there someplace. It’s not the numbers themselves that are problematic. I think you’re making the math intensional by considering propositional attitudes people have toward mathematical objects (i.e. assuming there are any such things at all–I leave those questions to Neil and the other mathematicians, myself).
Anyhow, in sum, I’d like to add you to the list of people on this thread whose posts have made very little sense to me lately. When I said I agreed with you I was agreeing that some of the remarks that others were making on extensionality were confused. But now I see your own views up close, they don’t make any sense to me either.
ETA: I should have said above that there’s no problem quantifying if we lose the quotation marks. Obviously, those can cause problems.
If the subformula being substituted for does not involve some kind of logical operation like equality or comparison* then its extension is a number. This also assumes of course that all the variables have assigned numbers through some definition (or by being bound without existential operators); the definition of extensional context needs closed formula, I believe.
It’s really no different than the standard “morning star” versus “evening star” examples. That substitution involves terms with the same extension, but that extension is not a truth value. Further, the two terms have different intensions, which is where Keith seems to be concentrating but which I say is irrelevant to extensional contexts.
(Of course, if we apply the extensional context definition to first order logic instead of English or math, then the only possible extensions are truth values, so that case can be confusing perhaps when we compare it to math or English).
———————
* or set and subset relations if we extend the domain of the variables in the forumlae from numbers to sets)
I take all formulae to involve some kind of logical operation like equality or comparison. The definition even specifically indicates that formulae have truth-values for extensions.
So read it that way and I think you’ll see what I was saying about necessity and contingency.
I think we agree on this, at any rate, Bruce. The criticisms of Quine on this thread have been confused. And I think Hylton would be in our court too!
ETA: One other thing I want to say is that I HOPE that definition of “extensional” that we’ve been talking about is widely accepted, because if it isn’t, it should be replaced by something simpler, that can be understood more easily. I haven’t criticized it based on the theory that it’s a consensus definition, but I kind of hate it. If you try substituting formulae and sub-formulae in there as instructed, it’s quite hard to tell what the hell is going on.
The video I linked above may help in explaining the distinction between intensional definitions and intensional contexts that I am trying to point out.
By Quine’s position, I am referring to his preference for first-order language to express scientific theories, partly because of its extensionality. I claim math shares that quality so it works for Quine.
I’ve seen other posts mentioning eg Quine on analytic versus synthetic. That issue is not what I am referring to.
Did you ask me that before? I must have missed it.
My point have been that math will serve Quine’s purpose of have a language without intensionality as his ideal theory language because math is extensional in the sense he wants it to be. *
I’m not trying to make any points about what it means for physics not to be expressed using math.
I don’t think contingency is relevant to the points I am making, if you are referring to it to say eg physics is empirical and hence its theories are contingent. That has no bearing that I can see on the extensionality of math formula as per the definition.
Physicists don’t use modality in the math used to express theories. Modality would introduce intensional contexts of course.
—————–
* That part of Quine is in the later chapters of W&O which we did not cover in QP. Of course, I am going mainly by Hylton’s construal of Quine.
BruceS,
I am hardly in a position to substitute my judgment for Quine’s on whether math is extensional. I leave that to keiths. My main point was that KN’s critique was mushroom-based.
Yes, whole formulas have the truth value in question for the definition. Just like whole sentences are used for the English version of intensional contexts (which BTW Quine used terms like “referential opacity” or “opaque contexts” for).
But the definition involves substitution for parts of the formula or sentence. Those substituted parts can have a non-truth value extension.
For math, the extension is a number. For English it is the common object in the worlds (“morning star” for “evening star”).
I’m still confused about why you think modality is involved. Are you applying modality to the definition itself or to the formulas or sentences that definition refers to? I don’t think it can apply to the latter .
Again, I was just trying to see what sorts of propositions are extensional pursuant to that definition. I believe math ones are, but physics ones aren’t.
That’s not clear to me. The parts are required to be formulae. A formula is NOT a singular term. (Hey! it’s not MY definition.)
Well, I’m used to “singular term” being used for natural language. I have not seen it used for math formula. So I don’t know whether the term even applies to math.
Let me give another specific example, step by step.
7+1 < 20 / 2
has the extension True. Subformula
20 /2
has the extension 10. So does the subformula
sqrt (100)
So by extensionality of math, I can substitute without changing truth value
7+1 < sqrt (100)
and the resulting whole formula is indeed still true.
To what I see as Keith’s concern: This does not depend on the intensions of “20/2” versus “sqrt(100)”, only their extensions.
I think we are saying the same thing which I would summarize by saying math cannot operate on meanings. Instead it operates directly on the extensions of terms. It ignores their intensions. In other words:
Math is a syntactic engine, not a semantic engine.
Sound familiar? It reminds me of “Brains are syntactic engines, not semantic engines”. I can get between the two using this line of argument.
1. Brains operate according to the laws of physics (assumption)
2. Physics uses math to express its laws (this nature of physics is assumed).
3. Math is extensional in the sense of the definition (as we agree).
4. Math is a syntactic engine, not a semantic engines (re-expressing 3).
5. Therefore, brains operate as syntactic engines, not semantic engines ((from 1, 2, 4).
I don’t really have a problem with what you’re saying there. It seems right to me. I just note that you’re not substituting a sub-formula there, just a mathematical expression that occurs in the single formula that you have put.
A formula is itself an equation, inequality, etc. So, what you’ve done is re-construe the definition in a manner that is easier to deal with. Again, I have no problem with that, just note that it’s not what the definition actually says. So, I wonder whether it is an accepted definition or just slopped onto the internet by somebody.
FWIW, I don’t think that’s valid.
Yes, fair enough. I guess I was using the word “formula” to apply to any math expression, including ones that evaluate to numbers, not truth values.
In my defense, I was taking a computer science perspective, since I am used to defining formula recursively by syntax so they can be analysed for evaluation. Now computer scientists tend to call such things expressions and sub-expressions, so my bad in switching to formula and sub-formula.
That confusion about terminology does not change my points about me, Keith, and Quine, and math, of course.
Well, it was offered more in fun than as something critical to the discussion, but I am interested in where you think it goes wrong, it case that is important to the discussion as I see it.
I question that step. Actually, I question some of the other steps, too.
The definition used in step 3 has to do with formulae, so is really about the use of logic within mathematics. But mathematics also include geometric thinking, so step 4 concludes too much.
BruceS,
I think the ‘uses math to express’ language is too weak to get anything good from it. It’s like the hand-wavy ‘done with’ we talked about above.
Suppose we put ‘chemistry’ instead of ‘physics’ in (1), ‘Chemistry uses test tubes to determine [something or other].’ in (2) and ‘test tubes are often made of glass’ in (3). Look at the fun we could have!
Neil Rickert,
I wasn’t considering the truth of any of the premises myself–only the validity of the argument.
I note that ‘entirely expressible in’ doesn’t work either, because if it did, physics would just BE a part of math. So you have to do something like make physics entirely expressible in terms of defined (physical?) entities and math. And I don’t know if that’s true or not. I take it Quine says it is? If so, i’ll take his word for it.
The issue of geometrical notation is possibly a valid point in general, although I am not sure if it applies to the mathematical formulae used to express the laws of physics.
I also think that extensional definition applies to the syntax of the formula with the truth value further applying once a model is chosen.
In particular, that definition of extentionalism does not involve any of the thinking that goes on in physicists in creating or understanding the physics that the formulae express. (And therein lies the source of some of the confusion regarding “done” IMHO).
walto,
The “=” operator applies to extensions, not intensions. To determine whether
is true, you first need to determine the extensions of the left-hand and right-hand sides of the equation. The extension of “5+7” is 12, and so is the extension of “10+2”. Substituting, the equation becomes
…which is true.
Ditto. Your implicit assumption is that the “=” operator must be applied to both sides of the equation before extensions are substituted for intensions, but that’s not how equations work. The equals sign applies to extensions, which is just another way of saying that equations are extensional contexts, as Bruce has been stressing.
My point is that without intensions, math becomes trivial.
and
are not very interesting. On the other hand,
…is fascinating because we are surprised that those two intensions have the same extension.
Who says that intensions have to create problems? Much of the time they don’t.
Maybe its an intension of the properties of 5 and 7, rather than of 12.
The problem with words is, when you use them too much, they become meaningless.
If soundness were a concern, I start by rejecting the reduction to physics of 1. But even if I accepted that reductionism and the argument could be modified to be sound with that reductionism, the conclusion that the “brain is a syntactic engine” would not be very interesting to me.
Semantics involves the mind, not the brain, at least when it comes to meaning. We already know from Putnam et al that meaning does not depend solely on the brain.*
———————————————
* OTOH, I am not so sure about phenomenology: perhaps it does depend solely on brain state. Pautz has empirical arguments against the view that phenomenology involves instantiated external properties, either directly (as in naive realism) or via represented properties (as in tracking intentionalism). (The empirical arguments say brain states and reported conscious experience correlate well, but that there are structural differences between reported experience and external properties). However, he still subscribes to intentionalism about phenomenology in the sense that uninstantiated properties are represented (I think, I’m still struggling with his modernized projectivism).
However, that latter argument relies on an intuition about the externality of experience, not on empirical arguments, AFAIK, and he does admit in one paper that one could choose instead to reject that intuition.
Quine does, See the video link I posted if you want the details. Not that I am claiming he is right to do so.
Bruce,
Really? He claims that intensions always cause problems?
When it comes to expressing the theories, physics is applied math.
Of course, determining which laws are both empirically adequate and also meet the pragmatic constraints of the scientific community is not part of math, it is science.
You don’t need to commit to whether science tells us anything about unobservable reality to do science. Whether science and its theories allow us to claim anything about the reality is not physics, it is philosophy.
ETA: My concern with the argument is that math is static, it does not do anything on its own. But the brain does. So perhaps you need so say that there is a person or computer involved to make the math do something. And then we are in the Chinese room.
Bruce,
And since that is my position also, I can’t figure out where you think the disagreement lies.
That’s hardly an unimportant point! Without intensions, math would be trivial. See my last comment to walto.
Bruce,
Perhaps it will help if I summarize my position. I hold that
1. Intension is essential to mathematics.
2. KN and Neil were therefore wrong to claim that
3. Equations are extensional contexts, but that in no way conflicts with #1 and #2 above.
keiths, to walto:
phoodoo:
Too much confusion to unpack there. You’re on your own, phoo.
The problem with words is that if you don’t understand them, they seem meaningless.
A quibble on this:
I agree that when it comes to well-formed formula in predicate logic, the formula will always have a truth value (assuming variables have a value or are bound by quantifiers).
I’m not sure if that extends to math expressions and subexpressions. If a well formed formula just means it is syntactically valid, then the extension need not be a truth value when it comes to math.
Now the definition I quoted for extensionality was for logical formulae, not math expressions. So I cheated a bit. But the definition seems readily extendable, and both Quine and SEP said math is extensional, so I figured I could get away with it.
Well, I could try ask you to unpack 1. But that would likely take us back to the morasse ot “doing” mathematics as opposed to “expressing” mathematics. And I don’t want to go there. So I won’t, at least not in the actual world.
It’s important sure. It just not relevant to the definition of extensionality and Quine’s needs. I had the impression from KN’s post that he thought it might be.
Anyway, at this point my brain hurts. Can syntactic engines do that?
(That’s a rhetorical question intended only for humor)
Bruce,
I already have. If you forbid intension, mathematics is reduced to the trivial. See this comment.
Bruce,
Those aren’t the only things being discussed here. We’re also talking about KN’s claim, backed by Neil, that
I disagree, and once again it’s because without intensional semantics, math is reduced to the trivial.
I generally agree with that, although I wouldn’t put it that extensions are being substituted for intensions. I’d simply say that the expression “5 + 7” and “6 + 6” (and maybe even “7+5”) don’t all mean the same thing– in some sense of “meaning.” A Fregean might say that the expressions have different intensions or senses. I don’t care for that way of looking at things myself, but there isn’t any substitution going on, in any case, I don’t think. The numbers are not intensions at any point.
I do think it’s right that whether there are also intensions floating around or not, only the extensions of these expressions can matter with respect to the truth-values, which is what I believe you and Bruce have been saying.
As previously noted, I don’t know what the “can be done” in
is supposed to mean, and I won’t try to guess.
ETA: Also “purely extensional semantics” seems to me to go pretty far if we allow the necessity operator. And, frankly, I don’t see why physics even requires that. (My sense is that those who think modal operators are ruled out by some principle of extensionality may be using the term “extensional” in a manner that’s so broad as to beg the question at issue.)
This seems to ascribe intension to computers.
I’m doubting that is what most people mean by “intension”.
walto,
Exactly! Different intensions, same extension. And since “=” applies to extensions,
…is true because both sides have the same extension — the number 12. The expressions (intensions) are different, however. Different intensions, same extension, and the latter is what matters when determining the truth of an equation.
Neil,
Computers routinely deal with intension. If I tell my text editor to perform an operation on all lines containing the word “Fregean”, I am specifying the set intensionally, not extensionally.
In other words, the set in question is all lines having the property P, where P is defined as ‘contains the word “Fregean”‘.
Neil,
I’m still curious about why you thought that fictionalism resolved the issue regarding intensionality in mathematics. Could you elaborate?
Only in extensional contexts, unfortunately.
keiths:
walto:
Why “unfortunately”? Your example…
…is an extensional context.
It is indeed. What’s unfortunate is that not all contexts are, and we can’t tell which ones are solely by seeing if there are = signs there.
walto,
It’s almost that easy. Besides scanning for equal signs, you can look to see whether an extensional interpretation makes more sense than the alternatives.
This…
…is pretty clearly not an extensional context, and it’s safe to say that the equal sign is not signifying equality here.
This, on the other hand…
…make sense when interpreted extensionally, but not otherwise.
The problem isn’t actually in stuff like
It’s in stuff like
and
Yet mathematicians manage to avoid such problems despite being immersed in a subject that is rife with intensions. They even deal successfully with numbers, like Chaitin’s Ω, that can only be specified intensionally.
keiths,
Well, you have certainly done a brilliant job of using words to clarify something.
What that something is, apparently no one but yourself will ever know.