I believe the “unreasonable effectiveness of mathematics in the natural sciences” can be explained based on the following idea. Physical systems prohibit logical contradiction, and hence, physical systems form just another kind of axiomatic, logical, and therefore mathematical system. To take a crude example, two different rocks cannot occupy the same point in space, due to logical contradiction. This allows the ability to mathematically talk about the rocks. Note that this example is definitively crude, since there are other things like bosons which actually can occupy the same position, but anyways, hopefully you get the idea.
What is the status of this argument in the philosophy of mathematics? Or general comments/references?
To take a crude example, two different rocks cannot occupy the same point in space, due to logical contradiction.
Except that....that isn’t a logical contradiction!
You have inadvertently demonstrated one of the best arguments for the study of mathematics: it stretches the imagination. The ability to imagine wild, exotic, crazy phenomena that seem to defy common sense—and thus, in particular, not to confuse common sense with logic—is crucial for anyone who seriously aspires to understand the world or solve unsolved problems.
When Albert Einstein said that imagination was more important than knowledge, this is surely what he meant.
I can see how that phrasing would strike you as being redundant or inaccurate. To try to clarify --
The rocks not occupying the same point in space is a logical contradiction in the following sense: If it wasn’t a logical contradiction, there wouldn’t be anything preventing it. You might claim this is a “physical” contradiction or a contradiction of “reality”, but I am attempting to identify this feature as a signature example of a sort of logic of reality.
Physical systems prohibit logical contradiction, and hence, physical systems form just another kind of axiomatic, logical, and therefore mathematical system.
The organization of facts into axioms, rules of inference, proofs, and theorems doesn’t seem to be an ontologically fundamental one. We superimpose this structure when we form mental models of things. That is, the logical structure of things exists in the map, not the territory.
You replied:
[T]he point I was offering was that the logical structure does exist in the territory, not just the map. Our maps are merely reflecting this property of the territory. The fundamental signature of this is the observation that physical systems, when viewed in a map which exists only as a re-representation (as opposed to an interpretation) amenable to logical analysis, are shown to prohibit logical contradiction. (For example, the two statements (if A, then B) and (if A, then not B) cannot both be true.)
Actually, they are both true if A itself is false. This is the import of the logical principle ex falso quodlibet.
But I take your point to be that certain logical statements (such as “A ⇒ ~~A”) are true of any actual physical system.
It is true that things are a certain way. They are not some other way. So, if a territory satisfies A, it follows that it does not satisfy ~A. And this is a fact about the territory. After all, the point of a map is to be something from which you can extract purported facts about the territory.
However, what is not in the territory is the delineation of its properties into axioms, on the one hand, and theorems, on the other. There are just the properties of the territory, all co-equal, none with logical priority. The territory just is the way it is.
For example, consider the statements “A” and “~~A”, where A is the application of some particular predicate to the territory. It is not as though there is one property or feature of the territory according to which it satisfies A, while there is some other property of the territory according to which it satisfies ~~A. That feature of the territory in virtue of which it satisfies A is exactly the same feature in virtue of which it satisfies ~~A.
In the logic, “A” and “~~A” are two distinct well-formed formulas, and it can be proven that one entails the other. But in the territory there are no two distinct features corresponding to these two wffs, so it’s not really sensible to speak of an entailment relationship in any nontrivial sense. The territory just is the way that the territory is, and this way, being the way that the territory is, is the way that the territory is. There is nothing more to be said with regard to the territory itself, qua logical system.
What about a tautology such as “A ⇒ ~~A”? Tautologies do give us true statements about the territory. But, importantly, such a statement is not true in virtue of any feature of the territory. The tautology would have been true no matter what features the territory had. There is nothing in the territory making “A ⇒ ~~A” be true of it. In contrast, there is something in the logical system making “A ⇒ ~~A” be a theorem in it — namely, certain axioms and rules of inference such that “A ⇒ ~~A” is derivable. (Some systems with different axioms or rules of inference would not have this wff as a theorem). This is another reason why the territory ought not to be thought of as a logical system of which the features are axioms or theorems.
Thank you for comment, and I hope this reply isn’t too long for you to read. I think your last sentence sums up your comment somewhat:
...the territory ought not to be thought of as a logical system of which the features are axioms or theorems.
In support of this, you mention:
What about a tautology such as “A ⇒ ~~A”? Tautologies do give us true statements about the territory. But, importantly, such a statement is not true in virtue of any feature of the territory. The tautology would have been true no matter what features the territory had. There is nothing in the territory making “A ⇒ ~~A” be true of it.
It seems like things are getting confused here. I take “A ⇒ ~~A” to be a necessary condition for proposition A to make sense. In order to make things concrete, let me use a real example. Say that proposition A is, “This particular rock weighs 1.5 pounds with uncertainty sigma.” This seems like a fairly reasonable, easily imaginable statement. Now clearly, A is simply a rendition or re-representation of the reality that is the physical system. In other words, proposition A only tells you what reality tells you by holding the rock in your hands, or throwing it through the air, or vaporizing it and measuring the amount of output energy. The only difference in this case is that the reality is encoded in human language.
For A to make sense, clearly “A ⇒ ~~A” must be true. For the rock to weigh 1.5 plus/minus sigma, it must not—not weigh 1.5 plus/minus sigma. That strikes me more or less a requirement imposed by human language, not so much a requirement of physical reality.
For this reason I think that your example of “A ⇒ ~~A” does not get to the heart of my point. My point is slightly different. Consider again the proposition “A true ⇒ (if A then B) OR (if A then not B)”. Take B as: “This rock is heavier than this pencil.” Now, assuming that the pencil does not lie in the weight range 1.5 plus/minus sigma, then this proposition must be true. And now, this statement is significantly more complicated than “A ⇒ ~~A”, and it implies that (under proper restrictions) you can make longer logical statements, and continuing further, statements which are no longer trivial and just a property of human language.
Side-note: I suppose these particular examples are all tautological so they don’t quite show the full richness of a logical system. However, it would be easy to make theorems, such as “if A AND C, then B” (where C could be specified similar to A or B.) Then we would see not only tautologies but also theorems and other propositions which are all encoded as we would expect from a typical logical system.
Now, the fact that this sort of statement works comes straight out of the territory. Our maps to A and B are merely re-representations of reality, and they are what reality is telling us, only encoded in human language. So we are seeing that reality appears to obey the same logical rules that we have come to expect from ordinary kinds of logical systems.
Now, I am not claiming that the physical systems (the territory) is somehow naturally encoding itself into these re-representations. Clearly, the human mind is at work in realizing these re-representations. But once these re-representations are realized, it really is the territory which takes on a logical structure.
So I am not claiming that the physical system is naturally a system of axioms and theorems and so on. My proposition is weaker and more generic, and only says that the physical system has a logical character. My real punchline, I suppose, is to say that this logical character of the re-representation is non-trivial. As you say, “Things are a certain way. They are not some other way.” But the way in which they are is logical. They are in a way which is the same way that logical statements are encoded. This is non-trivial because physical systems at the highest level just look like a huge collection of various and vague facts. We have no reason (a priori) to expect physical systems to map about in this way—but they do! And this I claim allows for math to be so effective in working with reality in general.
Consider again the proposition “A true ⇒ (if A then B) OR (if A then not B)”. Take B as: “This rock is heavier than this pencil.” Now, assuming that the pencil does not lie in the weight range 1.5 plus/minus sigma, then this proposition must be true. And now, this statement is significantly more complicated than “A ⇒ ~~A”, and it implies that (under proper restrictions) you can make longer logical statements, and continuing further, statements which are no longer trivial and just a property of human language.
I’m a little confused by this example. The proposition
A ⇒ (if A then B) OR (if A then not B)
is a logical tautology. It’s truth doesn’t depend on whether “the pencil does not lie in the weight range 1.5 plus/minus sigma”. In fact, just the consequent
(if A then B) OR (if A then not B)
by itself is a logical tautology. So, I have two questions:
(1) Is there a reason why you didn’t use just the consequent as your example? Is there a reason why it wouldn’t “get to the heart” of your point?
(2) Just to be perfectly clear, are you claiming that the truth of some tautologies, such as A ⇒ ~~A , is “trivial and just a property of human language”, while the truth of some other tautologies is not?
Sorry, I caught that myself earlier and added a sidenote, but you must have read before I finished:
Side-note: I suppose these particular examples are all tautological so they don’t quite show the full richness of a logical system. However, it would be easy to make theorems, such as “if A AND C, then B” (where C could be specified similar to A or B.) Then we would see not only tautologies but also theorems and other propositions which are all encoded as we would expect from a typical logical system.
Edit: Or, sorry, just to complete, in case you had read that—the tautology does depend on whether the pencil lies in the range of 1.5 plus/minus sigma. If the pencil lies in that range, we can’t say B or ~B.
In answer to (1.), I’m not using the consequent because you identified the fact that the consequent can imply anything by logical explosion. I was referring to the “A=>~A” example not getting to the heart of the point because that example is too simple to reveal anything of substance, as I subsequently discuss.
In answer to (2.), I am not claiming that some tautologies are “less true”. I am just roughly showing how there is a gradation from obvious tautologies to less obvious tautologies to tautologies which may not even be recognizable as tautologies, to theorems, and so on.
First, I, at least, am glad that you’re asking these questions. Even on purely selfish grounds, it’s giving me an opportunity to clarify my own thoughts to myself.
Now, I’m having a hard time understanding each of your paragraphs above.
Or, sorry, just to complete, in case you had read that—the tautology does depend on whether the pencil lies in the range of 1.5 plus/minus sigma. If the pencil lies in that range, we can’t say B or ~B.
B meant “This rock is heavier than this pencil.” So, “B or ~B” means “Either this rock is heavier than this pencil, or this rock is not heavier than this pencil.” Surely that is something that I can say truthfully regardless of where the pencil’s weight lies. So I don’t understand why you say that we can’t say “B or ~B” if the pencil’s weight lies in a certain range.
In answer to (1.), I’m not using the consequent because you identified the fact that the consequent can imply anything by logical explosion. I was referring to the “A=>~A” example not getting to the heart of the point because that example is too simple to reveal anything of substance, as I subsequently discuss.
I didn’t say that the consequent can imply anything “by logical explosion”. On the contrary, since the consequent is a tautology, it only implies TRUE things. Given any tautology T and false proposition P, the implication T ⇒ P is false.
More generally, I don’t understand the principle by which you seem to say that A ⇒ ~~A is “too simple”, while other tautologies are not. Or are you now saying that all tautologies are too simple, and that you want to focus attention on certain non-tautologies, like “if A AND C, then B” ?
In answer to (2.), I am not claiming that some tautologies are “less true”. I am just roughly showing how there is a gradation from obvious tautologies to less obvious tautologies to tautologies which may not even be recognizable as tautologies, to theorems, and so on.
But surely this is just a matter of our computational power, just as some arithmetic claims seem “obvious”, while others are beyond our power to verify with our most powerful computers in a reasonable amount of time. The collection of “obvious” arithmetic claims grows as our computational power grows. Similarly, the collection of “obvious” tautologies grows as our computational power grows. It doesn’t seem right to think of this “obviousness” as having anything to do with the territory. It seems entirely a property of how well we can work with our map.
B meant “This rock is heavier than this pencil.” So, “B or ~B” means “Either this rock is heavier than this pencil, or this rock is not heavier than this pencil.” Surely that is something that I can say truthfully regardless of where the pencil’s weight lies. So I don’t understand why you say that we can’t say “B or ~B” if the pencil’s weight lies in a certain range.
My idea was that the rock weighs 1.5 plus/minus sigma. If the pencil then weighs 1.5 plus/minus sigma, then you can’t compare their weights with absolute certainty. The difference in their weights is a statistical proposition; the presence of the sigma factor means that the pencil must weigh less than (1.5 minus sigma) or more than (1.5 plus sigma) for B or ~B to hold. But anyways, I might concede your point as I didn’t really intend this to be so technical.
I didn’t say that the consequent can imply anything “by logical explosion”. On the contrary, since the consequent is a tautology, it only implies TRUE things. Given any tautology T and false proposition P, the implication T ⇒ P is false.
Sorry, “logical explosion” is just a synonym for “ex falso quodiblet”, which you originally mentioned. You originally pointed out that the consequent can imply anything because of ex falso quodiblet, when A is not true. That wasn’t my intention, so I added the A true qualifier.
More generally, I don’t understand the principle by which you seem to say that A ⇒ ~~A is “too simple”, while other tautologies are not. Or are you now saying that all tautologies are too simple, and that you want to focus attention on certain non-tautologies, like “if A AND C, then B” ?
It initially seemed too simple for me, but maybe you are right. My original thinking was that “A ⇒ ~~A” seems to mean merely that a statement makes sense, whereas other propositions seem to have more meaning outside of that context. Also, the class of tautologies between different propositions seems to generalize the class of tautologies with a single proposition.
… It doesn’t seem right to think of this “obviousness” as having anything to do with the territory. It seems entirely a property of how well we can work with our map.
I hadn’t really thought about this, and I’m not sure how important it is to the argument, although it is an interesting point. Maybe we should come back to this if you think this is a key point. For the moment I am going to move to the other reply...
I guess my original idea (i.e., the idea I had in my very first question in the open thread) was that the physical systems can be phrased in the form of tautologies. Now, I don’t know enough about mathematical logic, but I guess my intuition was/is telling me that if you have a system which is completely described by tautologies, than by (hypothetically) fine-graining these tautologies to cover all options and then breaking the tautologies into alternative theorems, we have an entire “mathematical structure” (i.e., propositions and relations between propositions, based on logic) for the reality. And this structure would be consistent, because we had already shown that the tautologies could be formed consistently using the (hypothetically) available data. Then physics would work by seizing on these structures and attempting to figure out which theorems were true, refining the list of theorems down into results, and so on and so forth.
I’m beginning to worry I might lose the reader do to the impression I am “moving the goalpost” or something of that nature… If this appears to be the case, I apologize and just have to admit my ignorance. I wasn’t entirely sure what I was thinking about to start out with and that was really why I made my post. This is really helping me understand what I was thinking.
Tell me whether the following seems to capture the spirit of your observation:
Let C be the collection of all propositional formulas that are provably true in the propositional calculus whenever you assume that each of their atomic propositions are true. In other words, C contains exactly those formulas that get a “T” in the row of their truth-tables where all atomic propositions get a “T”.
Note that C contains all tautologies, but it also contains the formula A ⇒ B, because A ⇒ B is true when both A and B are true. However, C does not contain A ⇒ ~B, because this formula is false when both A and B are true.
Now consider some physical system S, and let T be the collection of all true assertions about S.
Note that T depends on the physical system that you are considering, but C does not. The elements of C depend only on the rules of the propositional calculus.
Maybe the observation that you are getting at is the following: For any actual physical system S, we have that T is closed under all of the formulas in C. That is, given f in C, and given A, B, . . . in T, we have that the proposition f(A, B, . . .) is also in T. This is remarkable, because T depends on S, while C does not.
This looks somewhat similar to what I was thinking and the attempt at formalization seems helpful. But it’s hard for me to be sure. It’s hard for me to understand the conceptual meaning and implications of it. What are your own thoughts on your formalization there?
I’ve also recently found something interesting where people denote the criterion of mathematical existence as freedom from contradiction. This can be found on pg. 5 of Tegmark here, attributed to Hilbert.
This looks disturbingly similar to my root idea and makes me want to do some reading on this stuff. I have been unknowingly claiming the criterion for physical existence is the same as that for mathematical existence.
This looks somewhat similar to what I was thinking and the attempt at formalization seems helpful. But it’s hard for me to be sure. It’s hard for me to understand the conceptual meaning and implications of it. What are your own thoughts on your formalization there?
I’m inclined to think that it doesn’t really show anything metaphysically significant. When we encode facts about S as propositions, we are conceptually slicing and dicing the-way-S-is into discrete features for our map of S. No matter how we had sliced up the-way-S-is, we would have gotten a collection of features encoded as proposition. Finer or coarser slicings would have given us more or less specific propositions (i.e., propositions that pick out minuter details).
When we put those propositions back together with propositional formulas, we are, in some sense, recombining some of the features to describe a finer or coarser fact about the system. The fact that T is closed under all the formulas in C just says that, when we slice up the-way-S-is, and then recombine some of the slices, what we get is just another slice of the-way-S-is. In other words, my remark about T and C is just part of what it means to pick out particular features of a physical system.
Though the word “tautology” is often used to refer to statements like (A v ~A), in mathematical logic any true statement is a tautology. Are you talking about the distinction between axioms and derived theorems in a formal system?
I’m not aware of any strong emphasis on this argument. It seems at first glance to be problematic at multiple levels.
One problem with your approach is that humans have evolved in a single, very well-behaved universe. So we have intuition both from instinct and from internalized experience that makes it very hard for us to tell what is actually a logical contradiction and what is not. Indeed, one of the reasons I suspect that so many people have issues with things like special and general relativity as well as quantum mechanics is that they can’t get over that these aspects of the universe don’t fit well with their intuitions.
Consider for a moment what a universe would look like where 1 + 1 did not equal 2. What would that look like? It isn’t clear to me that this is even a meaningful question. But that may be because these concepts are so ingrained in us that we can’t think without them. Thus, it may be that math works well for understanding the universe because humans have no other option. One could imagine us meeting an alien species that has some completely different but very effective way of understanding the universe that isn’t isomorphic to math at all.
Logical operation is quite well defined, with or without regards to human perception of that logic. The idea that logic may not be understood does not contradict the idea that an internal logic (may) underlie physical systems. (Note, maybe see my clarification below, here. )
Granted logic is somewhat mysterious and it is hard to imagine what a different kind of logic would look like. However, that is immaterial to my idea. The idea is just that you have signatures of illogic (e.g., both statements (a.) if A, then B, and (b.) if A, then not B, both true at the same time) which seem to be non-present in physical systems.
To take a crude example, two different rocks cannot occupy the same point in space, due to logical contradiction.
I’d say the real reason is the Pauli principle, which is a physical law not entailed by logic alone. I see no logical contradiction at all imagining two rocks that occupy the same place, like a 3D version of the two-dimensional picture you’d get by using two projectors to project their pictures onto the same point on a screen.
Around the turn of the last century, the logicists, like Frege and Russell, attempted to reduce all of mathematics to logic; to prove that all mathematical truths were logical truths. However, the systems they used (provably) failed, because they were inconsistent.
Furthermore, it seems likely that any attempt at logicism must fail. Firstly, any system of standard mathematics requires the existence of an infinite number of numbers, but modern logic generally has very weak ontological commitments: they only require the existence of a single object. For mathematics to be purely logical, it must be tautological—true in every possible world*, and yet any system of arithmetic will be false in a world with a finite number of elements.
Secondly, both attempts to treat numbers as objects (Frege) or concepts/classes (Russell) have problems. Frege’s awful arguments for numbers being objects notwithstanding, he has trouble with the Julius Caesar Objection; he can’t show that the number four isn’t Julius Caesar, because what this (abstract) object is is quite under-defined. Using classes for numbers might be worse; on both their systems, classes form a strict hierarchy, with a nth level classes falling under (n+1)th classes, and no other. Numbers are defined as being the concept which has all those concept’s whose elements are equinumerous; the class of all pairs, the class of all triples, etc. But because of the stratification, the class of all pairs of objects is different from the class of all pairs of first level classes, which is different from the class of all pairs of second level classes, and so on. As such, you have an infinite number of ’2’s, with no mathematical relations between them. Worse, you can’t count a set like {blue chair, red chair, truth, justice}, because it contains objects and concepts.
What seems more likely to me is that there are an infinite variety of mathematical structures, purely syntax without any semantic relevance to the physical world, and without ‘existence’ in any real sense, as a matter of induction we’ve realised that some can be interpreted in manners relevant to the external world. As evidence, consider the fact that different, mathematics are applicable in areas: probability theory here, complex integration here, addition here, geometry here...
Around the turn of the last century, the logicists, like Frege and Russell, attempted to reduce all of mathematics to logic; to prove that all mathematical truths were logical truths. However, the systems they used (provably) failed, because they were inconsistent.
No, that’s not right. Russell and Whitehead’s Principia Mathematica is the fullest statement of logicism, and its system was never proved inconsistent.
But because of the stratification, the class of all pairs of objects is different from the class of all pairs of first level classes, which is different from the class of all pairs of second level classes, and so on. As such, you have an infinite number of ’2’s, with no mathematical relations between them.
Here I’m less certain, but I’m pretty sure that that’s not right either. You would have relations among two such 2s, but those relations would be of a higher type than either 2. But, again, I’m definitely vaguer on how that would work.
Yes, sorry, I meant that Frege failed his system was inconsistent (though possibly not if you replace Basic Law 5 with Hume’s Law). Russell, on the other hand, simply runs into Incompleteness; you can’t prove all of mathematics from logic because you can’t prove it full stop.
You would have relations among two such 2s, but those relations would be of a higher type than either 2. But, again, I’m definitely vaguer on how that would work.
You’d have ’2’s of all cardinalities, so to have a relation between them, you would need to move into the uncountables—but then there are new pairs to be formed here… Essentially, you can reconstruct Russell’s original paradox, comparing the cardinality of the set with the cardinality of certain things that fall under it.
You could mitigate this but cutting short the recursion, and simply allowing the relation to hold between the first n levels of concepts or so., on pain of arbitrariness.
I’m curious as to the downvotes; was I off-topic, too long, or simply wrong?
Edit: And (if it’s acceptable to ask about other people’s downvotes) why was zero call downvoted?
Russell, on the other hand, simply runs into Incompleteness; you can’t prove all of mathematics from logic because you can’t prove it full stop.
That’s not a problem for logicism per se. Logicism isn’t really a claim about what it takes to prove mathematical claims. So it doesn’t fail if you can’t prove some mathematics by a certain means. Rather, logicism is a claim about what mathematical assertions mean. According to logicism, mathematical claims ultimately boil down to assertions about whether certain abstract relationships among predicates entail other abstract relationships among predicates, where this entailment holds completely regardless of the meaning of the predicates. That is, mathematical claims boil down to claims of pure logical entailment.
So, if you discover that your particular mathematical system is incomplete, then what you’ve really done is discover that you had missed some principles of logic. It’s as though you’d known that P ∧ P entails P, but you just hadn’t noticed that P ∨ P entails P as well.
(But you were right about why logicism ultimately failed to convince everyone: Mathematics seems to have ontological commitments, where pure logic does not.)
I’m curious as to the downvotes; was I off-topic, too long, or simply wrong? Edit: And (if it’s acceptable to ask about other people’s downvotes) why was zero call downvoted?
I didn’t downvote either comment. Your comment was probably downvoted because some readers considered its arguments to be wrong or unclear. zero call’s comment was probably downvoted because it smacks of the mind projection fallacy, especially here:
Physical systems prohibit logical contradiction, and hence, physical systems form just another kind of axiomatic, logical, and therefore mathematical system.
The organization of facts into axioms, rules of inference, proofs, and theorems doesn’t seem to be an ontologically fundamental one. We superimpose this structure when we form mental models of things. That is, the logical structure of things exists in the map, not the territory.
The organization of facts into axioms, rules of inference, proofs, and theorems doesn’t seem to be an ontologically fundamental one. We superimpose this structure when we form mental models of things. That is, the logical structure of things exists in the map, not the territory
I wish you would have made this last comment on the post directly, so that I could reply to that there. Anyways, the point I was offering was that the logical structure does exist in the territory, not just the map. Our maps are merely reflecting this property of the territory. The fundamental signature of this is the observation that physical systems, when viewed in a map which exists only as a re-representation or translation (as opposed to an interpretation) amenable to logical analysis, are shown to prohibit logical contradiction. (For example, the two statements (if A, then B) and (if A, then not B) cannot both be true, where A and B are statements in some re-representation of the physical system.)
I appreciate your comments but I’m having trouble seeing your point with regards to the idea. To reiterate, with regards to your last paragraph,
… that some can be interpreted in manners relevant to the external world …
I’m proposing that these interpretations work because the internal physical systems (the territory) obeys the same properties as consistent mathematical systems—see my comment to TM below.
There is a great deal of difference between it operating, in certain regards, on the same sort of rules (rules isomorphic to) mathematics, and mathematics being applicable because physics isn’t logically inconsistent. It’s not a logical contradiction to say that two points have the same position, nor to say that 2+2=1 (for the latter, consider arithmetic modulo 3). Nor can maths be deduced purely from logic; partly because logic doesn’t require the existence of more than one object.
Russell did try to deduce maths from logic plus some axioms about how the world worked—that there were an infinite number of things, etc., but the applicability of the maths is always going to be an empirical question.
I believe the “unreasonable effectiveness of mathematics in the natural sciences” can be explained based on the following idea. Physical systems prohibit logical contradiction, and hence, physical systems form just another kind of axiomatic, logical, and therefore mathematical system. To take a crude example, two different rocks cannot occupy the same point in space, due to logical contradiction. This allows the ability to mathematically talk about the rocks. Note that this example is definitively crude, since there are other things like bosons which actually can occupy the same position, but anyways, hopefully you get the idea.
What is the status of this argument in the philosophy of mathematics? Or general comments/references?
Except that....that isn’t a logical contradiction!
You have inadvertently demonstrated one of the best arguments for the study of mathematics: it stretches the imagination. The ability to imagine wild, exotic, crazy phenomena that seem to defy common sense—and thus, in particular, not to confuse common sense with logic—is crucial for anyone who seriously aspires to understand the world or solve unsolved problems.
When Albert Einstein said that imagination was more important than knowledge, this is surely what he meant.
I can see how that phrasing would strike you as being redundant or inaccurate. To try to clarify --
The rocks not occupying the same point in space is a logical contradiction in the following sense: If it wasn’t a logical contradiction, there wouldn’t be anything preventing it. You might claim this is a “physical” contradiction or a contradiction of “reality”, but I am attempting to identify this feature as a signature example of a sort of logic of reality.
In this comment, I wrote:
You replied:
Actually, they are both true if A itself is false. This is the import of the logical principle ex falso quodlibet.
But I take your point to be that certain logical statements (such as “A ⇒ ~~A”) are true of any actual physical system.
It is true that things are a certain way. They are not some other way. So, if a territory satisfies A, it follows that it does not satisfy ~A. And this is a fact about the territory. After all, the point of a map is to be something from which you can extract purported facts about the territory.
However, what is not in the territory is the delineation of its properties into axioms, on the one hand, and theorems, on the other. There are just the properties of the territory, all co-equal, none with logical priority. The territory just is the way it is.
For example, consider the statements “A” and “~~A”, where A is the application of some particular predicate to the territory. It is not as though there is one property or feature of the territory according to which it satisfies A, while there is some other property of the territory according to which it satisfies ~~A. That feature of the territory in virtue of which it satisfies A is exactly the same feature in virtue of which it satisfies ~~A.
In the logic, “A” and “~~A” are two distinct well-formed formulas, and it can be proven that one entails the other. But in the territory there are no two distinct features corresponding to these two wffs, so it’s not really sensible to speak of an entailment relationship in any nontrivial sense. The territory just is the way that the territory is, and this way, being the way that the territory is, is the way that the territory is. There is nothing more to be said with regard to the territory itself, qua logical system.
What about a tautology such as “A ⇒ ~~A”? Tautologies do give us true statements about the territory. But, importantly, such a statement is not true in virtue of any feature of the territory. The tautology would have been true no matter what features the territory had. There is nothing in the territory making “A ⇒ ~~A” be true of it. In contrast, there is something in the logical system making “A ⇒ ~~A” be a theorem in it — namely, certain axioms and rules of inference such that “A ⇒ ~~A” is derivable. (Some systems with different axioms or rules of inference would not have this wff as a theorem). This is another reason why the territory ought not to be thought of as a logical system of which the features are axioms or theorems.
Thank you for comment, and I hope this reply isn’t too long for you to read. I think your last sentence sums up your comment somewhat:
In support of this, you mention:
It seems like things are getting confused here. I take “A ⇒ ~~A” to be a necessary condition for proposition A to make sense. In order to make things concrete, let me use a real example. Say that proposition A is, “This particular rock weighs 1.5 pounds with uncertainty sigma.” This seems like a fairly reasonable, easily imaginable statement. Now clearly, A is simply a rendition or re-representation of the reality that is the physical system. In other words, proposition A only tells you what reality tells you by holding the rock in your hands, or throwing it through the air, or vaporizing it and measuring the amount of output energy. The only difference in this case is that the reality is encoded in human language.
For A to make sense, clearly “A ⇒ ~~A” must be true. For the rock to weigh 1.5 plus/minus sigma, it must not—not weigh 1.5 plus/minus sigma. That strikes me more or less a requirement imposed by human language, not so much a requirement of physical reality.
For this reason I think that your example of “A ⇒ ~~A” does not get to the heart of my point. My point is slightly different. Consider again the proposition “A true ⇒ (if A then B) OR (if A then not B)”. Take B as: “This rock is heavier than this pencil.” Now, assuming that the pencil does not lie in the weight range 1.5 plus/minus sigma, then this proposition must be true. And now, this statement is significantly more complicated than “A ⇒ ~~A”, and it implies that (under proper restrictions) you can make longer logical statements, and continuing further, statements which are no longer trivial and just a property of human language.
Side-note: I suppose these particular examples are all tautological so they don’t quite show the full richness of a logical system. However, it would be easy to make theorems, such as “if A AND C, then B” (where C could be specified similar to A or B.) Then we would see not only tautologies but also theorems and other propositions which are all encoded as we would expect from a typical logical system.
Now, the fact that this sort of statement works comes straight out of the territory. Our maps to A and B are merely re-representations of reality, and they are what reality is telling us, only encoded in human language. So we are seeing that reality appears to obey the same logical rules that we have come to expect from ordinary kinds of logical systems.
Now, I am not claiming that the physical systems (the territory) is somehow naturally encoding itself into these re-representations. Clearly, the human mind is at work in realizing these re-representations. But once these re-representations are realized, it really is the territory which takes on a logical structure.
So I am not claiming that the physical system is naturally a system of axioms and theorems and so on. My proposition is weaker and more generic, and only says that the physical system has a logical character. My real punchline, I suppose, is to say that this logical character of the re-representation is non-trivial. As you say, “Things are a certain way. They are not some other way.” But the way in which they are is logical. They are in a way which is the same way that logical statements are encoded. This is non-trivial because physical systems at the highest level just look like a huge collection of various and vague facts. We have no reason (a priori) to expect physical systems to map about in this way—but they do! And this I claim allows for math to be so effective in working with reality in general.
I’m a little confused by this example. The proposition
A ⇒ (if A then B) OR (if A then not B)
is a logical tautology. It’s truth doesn’t depend on whether “the pencil does not lie in the weight range 1.5 plus/minus sigma”. In fact, just the consequent
(if A then B) OR (if A then not B)
by itself is a logical tautology. So, I have two questions:
(1) Is there a reason why you didn’t use just the consequent as your example? Is there a reason why it wouldn’t “get to the heart” of your point?
(2) Just to be perfectly clear, are you claiming that the truth of some tautologies, such as A ⇒ ~~A , is “trivial and just a property of human language”, while the truth of some other tautologies is not?
Sorry, I caught that myself earlier and added a sidenote, but you must have read before I finished:
Edit: Or, sorry, just to complete, in case you had read that—the tautology does depend on whether the pencil lies in the range of 1.5 plus/minus sigma. If the pencil lies in that range, we can’t say B or ~B.
In answer to (1.), I’m not using the consequent because you identified the fact that the consequent can imply anything by logical explosion. I was referring to the “A=>~A” example not getting to the heart of the point because that example is too simple to reveal anything of substance, as I subsequently discuss.
In answer to (2.), I am not claiming that some tautologies are “less true”. I am just roughly showing how there is a gradation from obvious tautologies to less obvious tautologies to tautologies which may not even be recognizable as tautologies, to theorems, and so on.
First, I, at least, am glad that you’re asking these questions. Even on purely selfish grounds, it’s giving me an opportunity to clarify my own thoughts to myself.
Now, I’m having a hard time understanding each of your paragraphs above.
B meant “This rock is heavier than this pencil.” So, “B or ~B” means “Either this rock is heavier than this pencil, or this rock is not heavier than this pencil.” Surely that is something that I can say truthfully regardless of where the pencil’s weight lies. So I don’t understand why you say that we can’t say “B or ~B” if the pencil’s weight lies in a certain range.
I didn’t say that the consequent can imply anything “by logical explosion”. On the contrary, since the consequent is a tautology, it only implies TRUE things. Given any tautology T and false proposition P, the implication T ⇒ P is false.
More generally, I don’t understand the principle by which you seem to say that A ⇒ ~~A is “too simple”, while other tautologies are not. Or are you now saying that all tautologies are too simple, and that you want to focus attention on certain non-tautologies, like “if A AND C, then B” ?
But surely this is just a matter of our computational power, just as some arithmetic claims seem “obvious”, while others are beyond our power to verify with our most powerful computers in a reasonable amount of time. The collection of “obvious” arithmetic claims grows as our computational power grows. Similarly, the collection of “obvious” tautologies grows as our computational power grows. It doesn’t seem right to think of this “obviousness” as having anything to do with the territory. It seems entirely a property of how well we can work with our map.
My idea was that the rock weighs 1.5 plus/minus sigma. If the pencil then weighs 1.5 plus/minus sigma, then you can’t compare their weights with absolute certainty. The difference in their weights is a statistical proposition; the presence of the sigma factor means that the pencil must weigh less than (1.5 minus sigma) or more than (1.5 plus sigma) for B or ~B to hold. But anyways, I might concede your point as I didn’t really intend this to be so technical.
Sorry, “logical explosion” is just a synonym for “ex falso quodiblet”, which you originally mentioned. You originally pointed out that the consequent can imply anything because of ex falso quodiblet, when A is not true. That wasn’t my intention, so I added the A true qualifier.
It initially seemed too simple for me, but maybe you are right. My original thinking was that “A ⇒ ~~A” seems to mean merely that a statement makes sense, whereas other propositions seem to have more meaning outside of that context. Also, the class of tautologies between different propositions seems to generalize the class of tautologies with a single proposition.
I hadn’t really thought about this, and I’m not sure how important it is to the argument, although it is an interesting point. Maybe we should come back to this if you think this is a key point. For the moment I am going to move to the other reply...
Little note to self:
I guess my original idea (i.e., the idea I had in my very first question in the open thread) was that the physical systems can be phrased in the form of tautologies. Now, I don’t know enough about mathematical logic, but I guess my intuition was/is telling me that if you have a system which is completely described by tautologies, than by (hypothetically) fine-graining these tautologies to cover all options and then breaking the tautologies into alternative theorems, we have an entire “mathematical structure” (i.e., propositions and relations between propositions, based on logic) for the reality. And this structure would be consistent, because we had already shown that the tautologies could be formed consistently using the (hypothetically) available data. Then physics would work by seizing on these structures and attempting to figure out which theorems were true, refining the list of theorems down into results, and so on and so forth.
I’m beginning to worry I might lose the reader do to the impression I am “moving the goalpost” or something of that nature… If this appears to be the case, I apologize and just have to admit my ignorance. I wasn’t entirely sure what I was thinking about to start out with and that was really why I made my post. This is really helping me understand what I was thinking.
Tell me whether the following seems to capture the spirit of your observation:
Let C be the collection of all propositional formulas that are provably true in the propositional calculus whenever you assume that each of their atomic propositions are true. In other words, C contains exactly those formulas that get a “T” in the row of their truth-tables where all atomic propositions get a “T”.
Note that C contains all tautologies, but it also contains the formula A ⇒ B, because A ⇒ B is true when both A and B are true. However, C does not contain A ⇒ ~B, because this formula is false when both A and B are true.
Now consider some physical system S, and let T be the collection of all true assertions about S.
Note that T depends on the physical system that you are considering, but C does not. The elements of C depend only on the rules of the propositional calculus.
Maybe the observation that you are getting at is the following: For any actual physical system S, we have that T is closed under all of the formulas in C. That is, given f in C, and given A, B, . . . in T, we have that the proposition f(A, B, . . .) is also in T. This is remarkable, because T depends on S, while C does not.
Does that look like what you are trying to say?
This looks somewhat similar to what I was thinking and the attempt at formalization seems helpful. But it’s hard for me to be sure. It’s hard for me to understand the conceptual meaning and implications of it. What are your own thoughts on your formalization there?
I’ve also recently found something interesting where people denote the criterion of mathematical existence as freedom from contradiction. This can be found on pg. 5 of Tegmark here, attributed to Hilbert.
This looks disturbingly similar to my root idea and makes me want to do some reading on this stuff. I have been unknowingly claiming the criterion for physical existence is the same as that for mathematical existence.
I’m inclined to think that it doesn’t really show anything metaphysically significant. When we encode facts about S as propositions, we are conceptually slicing and dicing the-way-S-is into discrete features for our map of S. No matter how we had sliced up the-way-S-is, we would have gotten a collection of features encoded as proposition. Finer or coarser slicings would have given us more or less specific propositions (i.e., propositions that pick out minuter details).
When we put those propositions back together with propositional formulas, we are, in some sense, recombining some of the features to describe a finer or coarser fact about the system. The fact that T is closed under all the formulas in C just says that, when we slice up the-way-S-is, and then recombine some of the slices, what we get is just another slice of the-way-S-is. In other words, my remark about T and C is just part of what it means to pick out particular features of a physical system.
Though the word “tautology” is often used to refer to statements like (A v ~A), in mathematical logic any true statement is a tautology. Are you talking about the distinction between axioms and derived theorems in a formal system?
I’m not aware of any strong emphasis on this argument. It seems at first glance to be problematic at multiple levels.
One problem with your approach is that humans have evolved in a single, very well-behaved universe. So we have intuition both from instinct and from internalized experience that makes it very hard for us to tell what is actually a logical contradiction and what is not. Indeed, one of the reasons I suspect that so many people have issues with things like special and general relativity as well as quantum mechanics is that they can’t get over that these aspects of the universe don’t fit well with their intuitions.
Consider for a moment what a universe would look like where 1 + 1 did not equal 2. What would that look like? It isn’t clear to me that this is even a meaningful question. But that may be because these concepts are so ingrained in us that we can’t think without them. Thus, it may be that math works well for understanding the universe because humans have no other option. One could imagine us meeting an alien species that has some completely different but very effective way of understanding the universe that isn’t isomorphic to math at all.
Logical operation is quite well defined, with or without regards to human perception of that logic. The idea that logic may not be understood does not contradict the idea that an internal logic (may) underlie physical systems. (Note, maybe see my clarification below, here. )
Granted logic is somewhat mysterious and it is hard to imagine what a different kind of logic would look like. However, that is immaterial to my idea. The idea is just that you have signatures of illogic (e.g., both statements (a.) if A, then B, and (b.) if A, then not B, both true at the same time) which seem to be non-present in physical systems.
I’d say the real reason is the Pauli principle, which is a physical law not entailed by logic alone. I see no logical contradiction at all imagining two rocks that occupy the same place, like a 3D version of the two-dimensional picture you’d get by using two projectors to project their pictures onto the same point on a screen.
Around the turn of the last century, the logicists, like Frege and Russell, attempted to reduce all of mathematics to logic; to prove that all mathematical truths were logical truths. However, the systems they used (provably) failed, because they were inconsistent.
Furthermore, it seems likely that any attempt at logicism must fail. Firstly, any system of standard mathematics requires the existence of an infinite number of numbers, but modern logic generally has very weak ontological commitments: they only require the existence of a single object. For mathematics to be purely logical, it must be tautological—true in every possible world*, and yet any system of arithmetic will be false in a world with a finite number of elements.
Secondly, both attempts to treat numbers as objects (Frege) or concepts/classes (Russell) have problems. Frege’s awful arguments for numbers being objects notwithstanding, he has trouble with the Julius Caesar Objection; he can’t show that the number four isn’t Julius Caesar, because what this (abstract) object is is quite under-defined. Using classes for numbers might be worse; on both their systems, classes form a strict hierarchy, with a nth level classes falling under (n+1)th classes, and no other. Numbers are defined as being the concept which has all those concept’s whose elements are equinumerous; the class of all pairs, the class of all triples, etc. But because of the stratification, the class of all pairs of objects is different from the class of all pairs of first level classes, which is different from the class of all pairs of second level classes, and so on. As such, you have an infinite number of ’2’s, with no mathematical relations between them. Worse, you can’t count a set like {blue chair, red chair, truth, justice}, because it contains objects and concepts.
What seems more likely to me is that there are an infinite variety of mathematical structures, purely syntax without any semantic relevance to the physical world, and without ‘existence’ in any real sense, as a matter of induction we’ve realised that some can be interpreted in manners relevant to the external world. As evidence, consider the fact that different, mathematics are applicable in areas: probability theory here, complex integration here, addition here, geometry here...
*strictly speaking, true in every structure.
No, that’s not right. Russell and Whitehead’s Principia Mathematica is the fullest statement of logicism, and its system was never proved inconsistent.
Here I’m less certain, but I’m pretty sure that that’s not right either. You would have relations among two such 2s, but those relations would be of a higher type than either 2. But, again, I’m definitely vaguer on how that would work.
Yes, sorry, I meant that Frege failed his system was inconsistent (though possibly not if you replace Basic Law 5 with Hume’s Law). Russell, on the other hand, simply runs into Incompleteness; you can’t prove all of mathematics from logic because you can’t prove it full stop.
You’d have ’2’s of all cardinalities, so to have a relation between them, you would need to move into the uncountables—but then there are new pairs to be formed here… Essentially, you can reconstruct Russell’s original paradox, comparing the cardinality of the set with the cardinality of certain things that fall under it.
You could mitigate this but cutting short the recursion, and simply allowing the relation to hold between the first n levels of concepts or so., on pain of arbitrariness.
I’m curious as to the downvotes; was I off-topic, too long, or simply wrong? Edit: And (if it’s acceptable to ask about other people’s downvotes) why was zero call downvoted?
That’s not a problem for logicism per se. Logicism isn’t really a claim about what it takes to prove mathematical claims. So it doesn’t fail if you can’t prove some mathematics by a certain means. Rather, logicism is a claim about what mathematical assertions mean. According to logicism, mathematical claims ultimately boil down to assertions about whether certain abstract relationships among predicates entail other abstract relationships among predicates, where this entailment holds completely regardless of the meaning of the predicates. That is, mathematical claims boil down to claims of pure logical entailment.
So, if you discover that your particular mathematical system is incomplete, then what you’ve really done is discover that you had missed some principles of logic. It’s as though you’d known that P ∧ P entails P, but you just hadn’t noticed that P ∨ P entails P as well.
(But you were right about why logicism ultimately failed to convince everyone: Mathematics seems to have ontological commitments, where pure logic does not.)
I didn’t downvote either comment. Your comment was probably downvoted because some readers considered its arguments to be wrong or unclear. zero call’s comment was probably downvoted because it smacks of the mind projection fallacy, especially here:
The organization of facts into axioms, rules of inference, proofs, and theorems doesn’t seem to be an ontologically fundamental one. We superimpose this structure when we form mental models of things. That is, the logical structure of things exists in the map, not the territory.
I wish you would have made this last comment on the post directly, so that I could reply to that there. Anyways, the point I was offering was that the logical structure does exist in the territory, not just the map. Our maps are merely reflecting this property of the territory. The fundamental signature of this is the observation that physical systems, when viewed in a map which exists only as a re-representation or translation (as opposed to an interpretation) amenable to logical analysis, are shown to prohibit logical contradiction. (For example, the two statements (if A, then B) and (if A, then not B) cannot both be true, where A and B are statements in some re-representation of the physical system.)
I’ll move that part of my comment there, with my apologies.
That’s quite alright—thank you for your discussion.
I appreciate your comments but I’m having trouble seeing your point with regards to the idea. To reiterate, with regards to your last paragraph,
I’m proposing that these interpretations work because the internal physical systems (the territory) obeys the same properties as consistent mathematical systems—see my comment to TM below.
There is a great deal of difference between it operating, in certain regards, on the same sort of rules (rules isomorphic to) mathematics, and mathematics being applicable because physics isn’t logically inconsistent. It’s not a logical contradiction to say that two points have the same position, nor to say that 2+2=1 (for the latter, consider arithmetic modulo 3). Nor can maths be deduced purely from logic; partly because logic doesn’t require the existence of more than one object.
Russell did try to deduce maths from logic plus some axioms about how the world worked—that there were an infinite number of things, etc., but the applicability of the maths is always going to be an empirical question.