This is a response to a post that was made last May in this blog by Elliot and can be found here. I’m making this a post as opposed to a comment in an attempt to renew interest in this topic. I did make some comments on the original post, and while I don’t disagree with what I stated there, I don’t think I directly addressed the issues raised, I will do so now.
Any criticism of my own view would, of course, be welcome.
Elliot begins:
Deduction is a bad idea which Popper accepted.
The word deduction in the above sentence is not clear. It sets the bar of criticism too high because it’s not clear what it is we are to criticize. Note the post ends with “This is not an attack on logic in general.” But to me deduction and logic are nearly synonymous. But perhaps this is cleared up by what comes next:
The idea is that certain arguments are “deductively valid”.
This is incorrect or perhaps correct but too vague or ambiguous. Note the following argument:
All swans are white.
Alf is a swan.
So, Alf is white.
This basically consists of two premises and a conclusion. What allows me to move from the premises to the conclusion? Wait, that sounds like the kind of question a justificationist would ask, right? And perhaps that is the suggested problem? Letting this go for just a minute, let’s go ahead and ask this: What allows me to move from the premises to the conclusion? It’s an inference. The study of logic is the study of those inferences. Likewise with the study of deduction.
So I take Elliot to be saying something like this:
The study of which inferences are valid is a bad idea.
When Popper talks about deduction, he’s taking about the study of inferences. So either what I’ve stated just above as Elliot’s claim is correct, or the claim can’t be about Popper. That is, if Popper is wrong about deduction, it can only be that he is wrong about saying some inferences are valid and some are not. Why? Because Popper never put forth the idea that some arguments are valid unless he was specifically referring to the inferences therein. He does say some arguments are invalid, such as the argument that from many particulars we can get a universal. But even here, that’s because he sees the inference that would have to be accepted as invalid.
A valid argument is one such that if the premises are true then the conclusion MUST be true.
Again, the phrase “valid argument” is throwing me off. If a “valid argument” is one that only makes use of “valid inferences” then I think there are valid arguments. (Premises are not an issue via this definition.)
However, if “valid argument” means something else (like a good argument), then I can’t decide if I accept the term or not until I know more.
If we try to substitute in inference into the above definition this is what we would get:
A valid inference is one such that if the premises are true then the conclusion must be true.
That sounds awkward to me because it’s not clear in the above sentence where the inference fits into all this. But this does basically look like the definition of a valid inference, so I’ll leave it as it is for now.
Now, the problem of validity in regards to inferences is very similar to the problem of truth when it comes to empirical theories.
The problem which is raised by Elliot about deduction (which is the study of valid and invalid inferences) also exists when we talk about true statements, especially those universal in character:
All swans are white.
All swans must be white.
If I am a fallibilist, then how can I make this assertion? Doesn’t the must connote a certainty that a fallibilist must eschew?
If one is a fallibilist, then what gives one the right to make the claim that any universal statement is true? A universal statement says something like, if this is a swan, then it must be white. That’s the claim. (If it’s not, our ability to criticize the statement via testing is severely hampered.)
So how can a fallibilist make such a claim?
I think the response is that this is just a conjecture. But that’s all we are saying about the assignment of validity to an inference. It’s just a conjecture.
(If someone is nitpicking here, there is a related discussion about how we define truth, but perhaps that can be saved for another time.)
Deduction thus has an anti-fallibilist character because it seeks certain knowledge about the implications of premises.
When the term anti-fallibilist character is used, it’s as if we were talking about an attitude. But what we need are specific rules that we should follow or not follow. It’s as if Elliot were looking at the statement and saying, it sounds to me like people who want validity in the study of deduction want this validity because they want certainty. They shouldn’t want certainty, because that’s not part of being fallible. Therefore, they shouldn’t bother with the study of deduction which is only about validity (and we can’t be sure about it anyway).
Now, it may be that Elliot intended to say something else. But the methodology I’ve described above to me is frightening because I don’t want to be judged via how much certainty I have or not. To me that is a private affair. I’m sure at different points in the French Revolution, the Russian Revolution, and the Chinese Cultural Revolution, people whose very attitudes were under question could find themselves in scary predicaments. I’m sure that Elliot readily agrees we shouldn’t do this. So either he meant something else or he didn’t see his argument as leading in this direction.
So, taking that into account, let’s examine this a little more.
In the Logic of Scientific Discovery, Popper assiduously avoids this problem of psychologism. He noted that knowledge was objective in some sense, and that we learn when we find error. Therefore he set out to set up rules we should follow that would help us find this error. He specifically argued our psychologies need not be a part of this.
Compare:
1. You should love the people around you.
2. You shouldn’t drive over 55 miles per hour.
I don’t say that rule 1 is not important. But it’s hard to make progress arguing over rules like that. On the other hand with rule number 2, it’s much easier to argue with this, because what is expected via the rule is clearer.
To say something has an anti-fallibilist character is like saying something has too much hate in it. It’s hard to know what to do with a claim like that. I mean, yeah, I agree we should try and have a fallible attitude. That’s right. Popper agrees, too. But note, in the Logic of Scientific Discovery Popper actually formulates a lot of rules that are much more like my 2 above. He actually assiduously avoids rules like number 1. He saves that for places like chapter 24 of The Open Society and Its Enemies where he is making an almost emotional appeal for his version of rationalism (critical rationalism) after having delivered many, many stinging criticisms of other versions of rationalism. (When I say almost emotional, Popper doesn’t tell us how to feel, but he asks us to look at the consequences of other theories of rationality then to choose accordingly.)
Now another problem is that certainty itself, as in being certain, can refer to more than one definition. And these two are often confused:
A. “known or proved to be sure”
B. “assured in mind or action”
I see B as not a serious problem. I see as A as something we just don’t have. I’m happy to argue we don’t have certainty as per A above, but I feel extreme hesitancy as to judging others in regards to B.
Also, it’s one thing to argue a fallible attitude is a good idea; it’s another to judge another man’s heart and ask if he is fallible enough. It’s one thing to argue love is a good idea; it’s another thing to argue a person (or a statement) is too hateful.
Our methodology should avoid this wherever possible. Popper in the Logic of Scientific Discovery is exemplary here.
Deductivists believe that we can create valid arguments, and we can evaluate whether arguments are deductively valid or not.
Well, I believe we can use of the concept of validity in regards to inferences. We want valid inferences just like we want true empirical claims. If we conjecture an argument follows valid inferences, and if I also accept the premises of the argument, I would accept the argument.
I don’t think I’m being clear. I find the phrase “valid argument” ambiguous. I find the phrase “valid inference” very clear.
If all that is meant by “valid argument” is that the argument follows “valid inferences” then I know how to determine whether an argument is valid. But if by “valid argument” it also means my premises must be validated, this can’t be done, so there are no valid arguments in that case. But there would still be valid inferences.
Why is this important? Look at the following argument:
Alf is a non-white swan.
Therefore, not all swans are white.
There’s an inference in there, and if it isn’t made explicit, then no one can criticize it. (I mean potentially criticize, of course.) But note, the inference has the character of a universal theory. It has to be right every time, or guess what? If an exception is found, it won’t matter.
That is, it must always follow, or our ability to criticize the inference is seriously impaired.
Compare:
All swans are white.
Some swans are white.
How can some swans are white be tested?
Likewise an inference that we describe as valid must always apply. We don’t apply the term valid to an inference if it is only right sometimes.
Of course, we could use another word other than valid, but it’s the definition that must stay.
My main point is this:
Making our inferences explicit and then judging which are valid and not valid opens us to greater criticism not less. We might even very well do this because we don’t feel certain about our claims!
Here is an example of an invalid inference:
All men are mortal. Socrates is mortal. [inference] Socrates is a man.
Note my premises are actually true in this case, as well as my conclusion. How do I know that this inference is invalid then? I can find an exception. Here is it is:
All men are mortal. Spot is mortal. [inference] Spot is a man.
In fact, Spot is a dog. So we should not use this inference. We should not regard it as valid.
Here’s how Popper defines validity [bold is my own]:
“If an inference is valid then if the premises are all true, the conclusion must be true; that is, the truth of the premises (if they are all true) is invariably transmitted to the conclusion; and the falsity of the conclusion (if it is false) is invariably retransmitted to at least one of the premises.”
Unended Quest, page 165.
The “must” is there, but it’s there because otherwise we can’t criticize the inference. We need it in our definition. Once the inference is explicit, you can attempt to find an argument where it doesn’t work. Which would be an exception to it. We couldn’t both accept the exception and the validity of the inference. Especially note, that we can use the inference backwards, in a manner of speaking, to criticize the premise. This use of logic is extremely important to critical rationalism. It’s hard to know how criticism could be carried out effectively in many cases if we couldn’t make an appeal to valid inferences.
Here again is how Popper defines validity, this time in Objective Knowledge, page 304:
“I look upon logic as the theory of deduction or of derivability, or whatever one chooses to call it. Derivability or deduction involves, essentially, the transmission of truth and the retransmission of falsity: in a valid inference truth is transmitted from the premises to the conclusion. This can be used especially in so-called ‘proofs’. But falsity is also retransmitted from the conclusion to (at least) one of the premises, and this is used in disproofs or reputations, and especially in critical discussions.
“We have premises and a conclusion; and if we show that the conclusion is false, and assume that the inference is valid, we know that at least one of our premises must be false. This is how logic is constantly used in critical discussion, for in a critical discussion we attempt to show that something is not in order with some assertion. We attempt to show it; and we may not succeed: criticism may be validly answered by counter-criticism.”
Again, there’s no more problem with validity than there is with truth. These concepts go hand in hand. What is important is that we recognize both are conjectures.
They believe philosophers accurately do this dozens of times in their life. Validity isn’t an impossible ideal but is within our grasp. This implies that violating fallibility can be done routinely.
One thing here that I want to emphasize is this: knowledge is objective. People aren’t.
Okay, that’s wrong if read in the wrong way.
What I want to say is that knowledge has objective properties that help us overcome our own subjectivity. Deduction is the study of some of these properties. We might not see an error if all we were relying on is our own subjective impression of the situation, but if we look at the logic (especially if we write it down), we might very well catch the error.
This is a big part of Karl Popper’s philosophy. He made this most explicit in this three worlds metaphysics. As long as our claims are objective, we can explore the objectivity of that claim, much as we explore anything else, and we can make (meta) claims about similar categories of claims. We can make a claim that specific categories of claims always lead to certain conclusions because they share a particular inference. There is nothing wrong with this. Doing this opens us up to greater criticism, not less.
Unless the inference invariably transmits truth from premises to conclusion, we can’t test the inference. We can test the inference’s validity by trying to find a case where true premises don’t yield a true conclusion. If we find such a counter example, we’ll then reject the inference. We’ll call it invalid, just like we call a theory false.
We should not confuse this issue with that of certainty. Some people have a lot of certainty (as a feeling of assurance), others have little. But both are capable of acting to bring criticism to bear against their theories. The study of deductive logic is an essential tool here. Moreover, a person might use deductive logic because they are uncertain about their arguments, and making them explicit will help bring criticism to bear on them.
And one more thing, in the comment section, Elliot noted that:
[Added note: Specifically Elliot is referring below to mathematical proofs, and he is responding to a comment made by me, seven years ago, that had just been reproduced into the discussion via me. My comment was as follows, “I am wondering if the notion of proof in mathematics is just a relic of justificationism, and the term misleading …” Of course, I was only wondering about the use of the word, but not questioning the underlying possibility of achieving necessary truth. Note, Elliot’s remarks don’t specify whether we can achieve necessary truth one way or the other. I guess. Honestly, I do not know. In my opinion, they could easily be taken to say there is no necessary truth and likewise no validity. Anyway, if Elliot rejects validity as defined by Popper, and many others, it’s not clear what he regards mathematical proofs as achieving as they require axioms (which operate much like inferences, and need to be valid).]
[added:
> I am wondering if the notion of proof in mathematics is just a relic of justificationism, and the term misleading …
end added.]
FYI this topic is covered in _The Fabric of Reality_. It points out that proving is a physical process — it uses a pen, a hand, a paper, a brain to store memories, eyes, motion, etc — and so our proofs must depend on our understanding of the physical objects used. But pretty much no one even thinks our knowledge of physics is certain, so it’s kind of ridiculous they can think proofs performed with physical objects could be certain.
[Added note: According to Elliot the above is a misquote specifically in how the context is represented below. He has stated he wants it fixed. It seems sufficient to note this, rather than alter what has already been written. Follow this link to the original comment. Readers can read the context and decide about this themselves.]
[added note 2: Not because I felt it was necessary, but because as so as to err on the side of being fair, I’ve added yet another note above the quote.]
The context in which this quote was given was in a discussion of whether or not validity was important in the study of deduction. Elliot’s claim here, as I understand it, is that we are using theories about paper and pencils and so on when we write out a proof. Therefore, we should not expect to ever be able to attain validity. (But note the shift to the word certain in a discussion about validity. Why? Validity can’t be any more certain than truth.)
Two points. First, as we are only making conjectures this shouldn’t matter. When we test a theory by making some observations, we write down those observations. So whether we assign truth to the hypothesis we are testing or some counter hypothesis, our decision is guided by implicit theories about paper and pencils. So? If I write out a proof, it could be I erred because of something not related to the proof itself. But so long as my results are conjectural anyway, this shouldn’t matter.
In the Fabric of Reality it seems to me that David Deutsch is making the same point about proofs. As long as our knowledge about mathematica proofs is conjectural, we shouldn’t be overly concerned about this problem. As long as we don’t systematically avoid exceptions (something perhaps harder to do in math), eventually we should make progress. So he’s hardly arguing against the concept of validity. Like Popper, he draws comparisons between empirical studies and the study of math. In both cases exceptions are of great use.
The problem arises here of associating fallibilism with an attitude of certainty, I think. Again, we can feel certain about something, but because of our philosophical outlook, we might accept that this feeling of certainty could mislead us. Thus we can act via methodological rules to always seek out criticism.
Of course, if by certainty, it’s meant we know something true (or proved to be true), then, of course, we don’t have that. But that’s peripheral to validity. Valid inferences can’t prove anything to be true. An inference only transmits the truth to the conclusions, so it can’t be any stronger than the premises stated in any particular argument.
A fallibilist might criticize one man’s quest for certainty, but he need not necessarily be afraid of (ambiguous and subjective feelings of) certainty either, so long as he acts so as to try and find criticism. If that criticism exists, and we act to take it seriously, I don’t think that certainty (about the theory being criticized) will last long. In this sense, certainty (as a feeling of assurance) seems innocuous.
Two statements:
X. You’re too certain.
Y. You’re not acting to seek out criticism of your view.
Y is a more helpful statement. X is not nearly as much of a problem as the former (if we’re talking about a feeling of assurance.)
I have several notes on this topic with a lot of quotes. I will try to forward some of these to the comment section later in case some others might find them useful.
Popper on logic in Objective Knowledge, starting on page 304:
So in the empirical sciences logic is all about criticism. And the stronger our logic the more criticism that can be brought to bear on our theories. Making our inferences explicit enhances criticism.
From Unended Quest, starting at the bottom of page 165 (all of section 32 “Induction; Deduction; Objective truth” is a must read in the context of talking about validity, logic and so on.) If you would like a longer selection please contact me.
Note the way Popper shows the inference in his example to be false. He finds a counterexample. Validity is a conjecture, just as truth is. It is not self-evident, based on intuition, or even gained via special access to Plato’s world of forms. Rather it is an objective property of knowledge that can be studied the way we study anything else, via conjecture and refutation.
On Popper’s non-psychological approach, from page 80 and 81 of The Logic of Scientific Discovery:
In the above passage you already see hints of world three.
Again, above, more hints of Popper’s later metaphysical concept of three worlds.
Note the very strong analogy above between how we do proofs and how we test our theories. Conjectures and Refutations. Did we really need Lakatos? 😉
I guess Popper further amended this later to talk about critical preferences.
I think David Deutsch is very good on the topic of mathematics and logic. He discusses this in chapter 10 of his book, _The Fabric of Reality_.
There are several things that are worth quoting and discussing there. However, I just want to choose something short right now:
Just as there are in many places in that chapter, there is parallel here to what Popper is saying. Look at the second to last quoted paragraph I give for _The Logic of Scientific Discovery_. What Popper expresses there and Deutsch here, seems nearly identical to me.
Just like Popper’s W1 is made up of physical reality, W3 objects can also be studied. In both case we use conjectures about what the properties of these objects are, however, the way we test these theories is a little different for each. There’s an irony in that W3 objects are (I guess, and Popper suggests) manmade, yet we do not at all fully understand them.
Also, even if a W3 object is that of a false theory (about W1), it still has independent properties we can study.
This matter of psychologism seems more and more important to me. Popper spent relatively little space in LoSD on the issue, and even his fans seem to have skipped over it rather uncomprehendingly. The perpetual disagreements and misunderstanding regarding induction seems to turn almost entirely on the fact that most inductivists take psychologism for granted, and can sarcely imagine approaching issues of epistemology or methodology any other way. You can read some of my recent comments on this blog on the matter–they echo much of what you have written here.
> The study of which inferences are valid is a bad idea.
Studying inferences is fine.
This is all way too long. I think it says you don’t understand my short post, and you’ve made various lengthy guesses instead of asking. If you would like to discuss, could you pick one issue to start with?
Matt,
Please be more careful with misquoting and edit the above post.
> > FYI this topic is covered in _The Fabric of Reality_.
> The context in which this quote was given was in a discussion of whether or not validity was important in the study of deduction
That is false. The context, which I helpfully quoted immediately preceding the text starting “FYI”, was:
> > > I am wondering if the notion of proof in mathematics is just a relic of justificationism, and the term misleading …
I replied to that. “this topic” refers to “proof in mathematics”. That’s the context of that comment which I even made explicit.
You’re welcome to break the post down and criticize only portions of it.
The above is an ambiguous response.
The word validity is not important. The definition is.
Popper’s definition is fine here:
“If an inference is valid then if the premises are all true, the conclusion must be true; that is, the truth of the premises (if they are all true) is invariably transmitted to the conclusion; and the falsity of the conclusion (if it is false) is invariably retransmitted to at least one of the premises.”
We only study inferences if we are interested in validity. At least, if we are still only talking about deduction.
The notion of validity as defined above has not been successfully criticized — or even criticized at all.
OK, you want to talk about validity. I will make an argument, step by step.
Now, answer carefully because I’m going to be extremely pedantic if you say anything wrong, and if you don’t want to discuss at high precision I will stop discussing (b/c I don’t think my point will be understood otherwise).
What is an example of a true statement?
Good post!
This isn’t an inquisition. It would be useful if you could make a claim and/or criticize a claim. If examples are needed I suggest you furnish them yourself.
Indeed, I wrote in the post:
“If someone is nitpicking here, there is a related discussion about how we define truth, but perhaps that can be saved for another time.”
It seems you want to have that discussion. Okay. Have at it, how do you define truth? That’s actually worthy of an entire blog entry.
If you won’t write a short statement of your position (as I did for mine), ask a specific question about what I wrote, answer my questions about your position which are part of an argument, or even try to be precise, then what do you want? It seems that you don’t want to discuss.
> It would be useful if you could make a claim and/or criticize a claim.
Some criticisms have more than one step. As I told you, I was making an argument. I was doing exactly what you ask — and I explicitly said I was — but you refuse to participate.
I’m content with what I’ve stated, Elliot.
“It seems that you don’t want to discuss.”
Note you see the problem here as a willingness to discuss or not discuss an issue. And you’re passing judgement on me based on what you conjecture is my unwillingness. But this methodology is precisely what I criticized in my post.
It’s like this, I presented some claims. Among these claims was some criticism of claims you had made. (But it didn’t matter who had made those claims, they could have been made by anyone.)
Your response is not to criticize the claims I made (except to say they are too long and that they are in error), but instead to criticize me, and to suggest that if you can’t elicit certain responses in some way from me, I’m at fault in some way.
But note, my claims could have been made by anyone. So it should be those you are concerned with, not me personally. If the claims I wrote are of no interest to you, then surely that is more a concern of yours than of mine, is it not?
when will the misquote be fixed?
I’ve linked to your comments in the post itself, and suggested people review your comments in this regards.
Not good enough Matt. I explained the problem with your quoting and you haven’t even replied, disagreed, agreed, or anything. It’s still a problem. This is now like misquoting someone in a book with a footnote saying the right thing. If you will not treat me with basic respect and fix the misquote, then remove me from the blog as a contributor and don’t ever ask me for anything in the future.
Also you misquoted me again, which you also need to fix. You said that I said “the above” is a misquote. But if you actually read my claim, it referred to a combination of the above and the below (explicitly with quoting).
Misquoting someone’s claim that you misquoted them is really careless.
I’ve adjusted the note slightly.
If you would like me to adjust your status from that of someone who can post blog entries to merely a member who post comments, I will of course comply. If you would like me to delete your membership entirely, then of course, I will comply. (And your ability to comment as a non-member will in no way be restricted.) Whatever you wish. Just let me know.
Of course, you would be welcome back at any time in the future.
It’s not acceptable to misquote anyone, let alone contributors. Yes remove me as a contributor.
This is too confusing for me because I have not engaged with the details of the discsussion.
However I have no reason to think that Matt is acting in bad faith and the simplest resolution would be, or would have been, for Elliot to restate his position vis a vis the point at issue and let Matt respond to that position again.
Rafe, just for the record, as per Elliot’s request, he has been removed as a contributor. He is welcome back should he change his mind in the future.
Matt I doubt that you are acting in bad faith, but you misquoted Elliot and you should fix the misquote. The original comment reads as follows:
Your use of the quote is a misquote because Elliot was referring specifically to mathematical proofs, rather than deductive arguments. You write “And one more thing, in the comment section, Elliot noted that:” before quoting the second paragraph quoted above and this follows on from a discussion of deduction and so it gives a misleading impression of what Elliot was discussing. If you want to correct the misquote you should either put the first paragraph of the quote in, or write “And one more thing, in the comment section, in the context of a discussion of whether mathematical proofs are justificationist, Elliot noted that:”.
In response to Alan:
I’ve added more text above the quote.
Note this sentence, “It points out that proving is a physical process — it uses a pen, a hand, a paper, a brain to store memories, eyes, motion, etc — and so our proofs must depend on our understanding of the physical objects used. ”
Is this the claim _The Fabric of Reality_ actually makes?
I read the above as implying that necessary truth is dependent on the laws of physics.
Premise: Proving is a physical process.
Premise: Physical processes make use of the laws of physics.
Conclusion: Proving must depend on the laws of physics.
Is this correct? I think not.
It implies as best I can determine that necessary truth is dependent on the laws of physics. I do not think that’s what Elliot wants to say.
I think he meant something else. The ambiguity is in the word proving.
As Alan has commented on this, I would be very interested if he wished to offer an opinion on this.
Of course, I don’t think that David Deutsch regards necessary truth as being dependent on the laws of physics. That idea, which I hope, no one intends to suggest is regarded by me as silly.
I just edited the added note above the quote in question in the original entry to reflect the comment made just above this one.
The relevant part of FoR is Chapter 10: The Nature of Mathematics. I haven’t read it for a while but I think the argument runs a bit like this. Mathematics is about necessary truths: that is, statements that would be true regardless of the laws of physics. For example, there are infinitely many primes and even if we only ever use a finite number of primes all of the others would still exist. Now as a matter of fact once somebody proposed the definition of a prime it was true that primes have this property and would still have it even if nobody discovered it. So mathematical entities have properties that are independent of our knowledge of those properties.
When you write down a proof you use physical objects such as pens and paper, or a computer, or your brain. You are making the conjecture that in some relevant respects the behaviour of those physical objects is the same as the behaviour of some set of abstract mathematical entities that you are interested in. If you were writing down a proof and the marks on the paper changed at each step but your memory also changed so that you didn’t notice the change then you might think you have proved that 1+1=3. I don’t think that this happens, but you get the idea. Furthermore, in the future we may do computations using quantum computers whose results we could not check ourselves because doing so would take more matter than is available in the entire universe. Proof theory, the theory of what can be proved, has usually been regarded as a branch of mathematics, but it is actually a branch of physics. What can be ‘proved’ in mathematics – that is, what we can know about abstract mathematical entities – is precisely the set of truths about those mathematical entities that can instantiated in physical objects. There is nothing else about the set of provable mathematical truths that picks them out from the much larger set of unprovable mathematical truths.
See also
http://www.qubit.org/people/david/structure/Documents//Research%20Papers/PPQT.pdf
http://www.qubit.org/people/david/structure/Documents/Research%20Papers/ItFromQubit.pdf
As a side note, I don’t think you should say that ideas are silly. This is not a substantive criticism that points out a specific feature of those ideas that does not make sense. Saying some idea is silly only appeals to people who already agree with you, and may put off the very people who need to have their ‘silly’ ideas criticised. I admit that I often haven’t resisted this temptation in the past, but I now think that it’s an important rule.
Alan,
It’s not clear to me why you are giving me David Deutsch’s views, but I will comment on what you say.
I expressed my own opinion on David Deutsch’s views in _the Fabric of Reality_ in the actual entry. Do you agree or disagree with what I stated there?
Again, please look at this sentence:
“It points out that proving is a physical process — it uses a pen, a hand, a paper, a brain to store memories, eyes, motion, etc — and so our proofs must depend on our understanding of the physical objects used. ”
It is not people who prove mathematical conclusions. It is the proof.
Therefore proving mathematical conclusions is not dependent on the laws of physics.
The correctness of any proof exists independent of what you or I or anyone else says or does about it. Unless you presume some type of de facto reductionism, the correctness of any proof exists independent of what the laws of physics say or do about it as well.
There’s no dependency here as far as proving.
Problems and their solutions can exist independent of any one person. So can proofs.
Now, I am not arguing that David Deutsch says otherwise. I have read chapter 10 of _the Fabric of Reality_ and I am still studying it. There are some areas that I’m currently thinking about, but overall, so far, I guess what he says sounds okay. I don’t think it contradicts anything I’m saying here. If you think it does, I’d like to know about it.
In regards to what you write about David Deutsch ideas, you state the following:
“What can be ‘proved’ in mathematics – that is, what we can know about abstract mathematical entities – is precisely the set of truths about those mathematical entities that can instantiated in physical objects. There is nothing else about the set of provable mathematical truths that picks them out from the much larger set of unprovable mathematical truths.”
Note the phrase:
“What can be ‘proved’ in mathematics”
Who is proving what to whom? Again it’s the proof that does the proving. Do you disagree?
Now, note what follows:
“what we can know”
So you are making an argument about the limits of knowledge. That’s fine, but I don’t see that as being related to what is being discussed. Do you disagree?
By the way, I would be interested if you could comment on what this discussion is primarily about:
I believe that deduction is the study of inferences. I believe that when we study inferences we want to know which are valid and which are not. I’ve provided Popper’s definition of validity and agree with it. Do you have any criticisms of this view? Do you feel the study of deduction is somehow misguided?
I agree that there are deductively valid inferences – that is, inferences in which if the premises are true then the conclusion is true. I don’t see any problem with studying deductively valid inferences. Nor do I see any problem with the notion that an inference is either valid or invalid independent of our knowledge about those inferences.
We make and test conjectures about which inferences are deductively valid. The limits of our ability to do this are set by the laws of physics.