Science is just a re-branding of logic

SophistiCat

Doesn't the difference entirely rest upon the normative and hence subjective context by which we judge behaviour to be future-anticipating? — sime

Of course warrant is normative. How can you say that it is both normative and vacuous? That seems contradictory.

Janus

I think that is the mistake you are making, and that Hume also made; is imagining that there could possibly be a logical reason — Janus

Hume imagined no such thing. On the contrary he pointed out that there couldn't be a logical reason, or at least (being a fairly humble fellow) that he had no hope of ever finding such a reason. — andrewk

I probably did not express that every well. What I meant to say is that Hume's mistake consists in imagining that it ever should have been thought that there could be purely logical reasons justifying inductive reasoning, and to claim it as an interesting insight that there are not such reasons. In other words it just seems silly to think that it could ever have turned out that not to believe in induction would involve a contradiction. How could it ever be thought, for example, that it would be self-contradictory to believe that it was possible the Sun would not rise tomorrow?

The point I find most inconsistent in Hume's position is that, although he claims we have no reason to believe in the regularities we observe constantly in nature, he seems to think there is good reason to believe that humans and other creatures habitually expect them. Such habitual expectation would itself be a regularity, so Hume's argument fails according to its own principles.

Returning to the example of the Sun's rising; it seems that there is actually a deductive principle at work, but that it relies on a premise which itself cannot be deductively proven (which is really quite trivial since such is the case for all deductive reasoning). Taking the Sun's rising as an exemplar of natural invariance, we can deduce that 'if nature is governed by necessary laws, then then the Sun will rise tomorrow unless some other unforeseen lawful event prevents it from doing so'. Or, taking Peirce's alternative, if we adopt the premise that "nature takes habits", we can deduce that it is most likely that the Sun will rise tomorrow, unless some greater unforeseen habit of nature intervenes.

If Hume's argument were taken to its logical conclusion then it would result in Meillassoux's radical contingency, and we could not even trust in our own memories of, or documents recording, what happened in the past, let alone our expectations of what is to come. All our discourse would then be thoroughly undermined and we would not be capable of saying anything sensible about anything at all.

apokrisis

Or, taking Peirce's alternative, if we adopt the premise that "nature takes habits", we can deduce that it is most likely that the Sun will rise tomorrow, unless some greater unforeseen habit of nature intervenes. — Janus

This is important as Peirce is giving an actual reason for why induction is something that strengthens with time. A constraints-based view of the world says induction should become ever more reliable because a reasonable habit will keep growing stronger in being reinforced by its own success.

That is how science works. Our conviction strengthens as a belief survives challenge to its applicability.

And that is how the solar system works. In its early days, the sun came up everyday on the earth in a more unreliable manner. It took a while for a ball of debris to even accumulate into a planet. The early solar system was fraught with broken up junk that could have smashed into and derailed the earth, ending any nascent habit of a daily dawning of the sun.

But over time, the solar system got cleared up of all the junk, all the chaos, and settled down into a long-term groove. The inductive grounds of a belief became ever firmer.

So while there is nothing absolute to warrant that the past predicts the future, a constraints-based view of causality makes it deductively reasonable that regularity develops over time. Habits want to emerge. Order predicts not just order, but increasing order. Constraints develop a weight that make it increasingly hard for individual accidents to derail.

So the principle of induction is - as Peirce put it - about the taking of habits.

The Humean view arises from imagining a reality without real interactions. It is a Newtonian paradigm where everything reduces to local accidents - random collisions. Of course, in a world imagined like that, you would expect there to be no gathering history, no developing state of generalised coherence. If everything is imagined as fundamentally random and memory-less, then of course the deductive consequence - Hume's argument - is that even the laws of nature might change for no reason at any time.

But once you have a metaphysics which can take account of interactions - see how that requires a generalised coherence to emerge just due to "randomness" - then you will deduce something quite different about nature. You will have a different model-theoretic view to test by observation.

So really Hume is advancing a metaphysics-based hypothesis - and one that is believed due to Newtonian science. That is what gives it any credence it might have.

However, the "shock" is that this Newtonian causality just isn't what we observe in nature - on the whole. Instead we see a world where interactions result in a generalised state of coherence. Constraints or habits inevitably - and logically! (we can do the maths of self-organisation!) - must emerge to bring predictable and increasing order to their "worlds".

Hume had the right argument for the wrong metaphysics. And physics has since moved on as well.

Gosh it's like someome here has never read Hume before. — StreetlightX

Ah. Hear the plaintiff squeak of someone who has never stopped to truly consider what Humean doubt is about. Such a big difference between reading about something and thinking about something.

Janus

↪apokrisis

That's a nice elaboration of what is entailed by the Treatise of Humean Nature. In his ambition to do for human nature what he thought Newton had done for Nature in general he seems to have fallen into a deeply contradictory position given that, without accepting the very regularities his position denied, no such ambition could be fulfilled. Hume seems to have both rejected, and yet put his faith in, the determining power of regularity. :)

andrewk

Hume's mistake consists in imagining that it ever should have been thought that there could be purely logical reasons justifying inductive reasoning, and to claim it as an interesting insight that there are not such reasons — Janus

My recollection is that Hume was not imagining this himself, but rather writing in response to Rationalists who not only imagined it but believed it possible. It sounds like you are agreeing with Hume that it was not.

he seems to think there is good reason to believe that humans and other creatures habitually expect them — Janus

Are you sure he wasn't observing that humans always have habitually expected them, which is past tense, and doesn't need to use induction.

If Hume's argument were taken to its logical conclusion .... All our discourse would then be thoroughly undermined and we would not be capable of saying anything sensible about anything at all.

No, because we can make the discourse perfectly well just by accepting the principle of induction without insisting on a warrant for it. Remember, Hume didn't say we shouldn't use induction, but rather that it seemed to him to be futile to search for a warrant for it.

We can draw a parallel between Hume and Godel.

In the early 20th century mathematicians, led by Hilbert, were engaged in a program of proving the soundness of mathematics. Godel proved that that was impossible. Did that mean that Godel claimed we shouldn't use mathematics? Of course not! He thought we should, but just that we should not waste our time trying to prove its foundations were sound.

Similarly, the Rationalists in the 16-18th century were trying to prove the soundness of reason and scientific methods. Hume showed that that was impossible. Did that mean that Hume claimed we shouldn't use reason? Of course not! He thought we should, but just that we should not waste our time trying to prove its foundations were sound.

sime

Of course warrant is normative. How can you say that it is both normative and vacuous? That seems contradictory. — SophistiCat

Sorry, i meant warrant being epistemologically vacuous.

Janus

My recollection is that Hume was not imagining this himself, but rather writing in response to Rationalists who not only imagined it but believed it possible. It sounds like you are agreeing with Hume that it was not. — andrewk

Can you cite a statement by any rationalist that says it would be logically contradictory for things not to be as they are, or not to be in the future as they have in the past?

It's true that rationalist philosophers such as Spinoza and Leibniz believed that things are necessarily as they are; but this is not a logical necessity, but rather in accordance with what they thought of as the necessity of the Divine Nature. It is a complicated story but it has nothing to do with pure logical necessity as in the law of non-contradiction.

Are you sure he wasn't observing that humans always have habitually expected them, which is past tense, and doesn't need to use induction. — andrewk

What so his argument only applied to the past then? Basing his account of human nature on past observations of regularity is no different, in principle, than basing an account of non-human nature on past observations of regularity. In any case if there is really no regularity in nature then memory itself could not be counted as reliable, which would render even his account of the past untenable.

No, because we can make the discourse perfectly well just by accepting the principle of induction without insisting on a warrant for it. Remember, Hume didn't say we shouldn't use induction, but rather that it seemed to him to be futile to search for a warrant for it. — andrewk

The argument is only over whether induction is rationally warranted. Hume says it isn't at all, and this entails that his own arguments are not rationally warranted either.

We can draw a parallel between Hume and Godel. — andrewk

That axioms are not proveable does not entail that they are not rationally warranted. They are rationally warranted because without them there can be no discourse. The irrational demand for absolute proof is the whole source of these kinds of humean errors of thought.

andrewk

↪Janus

The argument is only over whether induction is rationally warranted. Hume says it isn't at all, and this entails that his own arguments are not rationally warranted either.

which, being a pragmatist, doesn't bother him. He just assumes the principle of induction as an axiom, and then any arguments he makes are conditionally warranted based on acceptance of that axiom, which is all he, or any sensible pragmatist, wants.

The point is that one has to adopt that principle as an unfounded axiom, and arguments that one can somehow 'prove' the axiom are unneeded and unsound..

Janus

↪andrewk

My real complaint, though, is that Hume paints our acceptance of induction as being merely a matter of habit, and not in any way rationally justified. Also, I would not agree with classing Hume as a pragmatist. Perhaps you just meant that he was a pragmatic thinker? Otherwise we seem to be in agreement.

apokrisis

which, being a pragmatist, doesn't bother him. — andrewk

But Hume represents the nominalist turn of thought. He was not a pragmatist in the sense of arguing for the reality of the general or universal. He was an atomist in regards to empirical sense data. So his epistemology reflects a particular brand of metaphysics.

MindForged

↪Janus

That axioms are not proveable does not entail that they are not rationally warranted. They are rationally warranted because without them there can be no discourse. The irrational demand for absolute proof is the whole source of these kinds of humean errors of thought.

I think I was mostly with you until you said this (depending on what you meant). If by this you meant a particular set of axioms are rationally warranted because without them discourse is impossible, I would find that dubious (people disagree about what axioms should be adopted in math and logic, and they do so intelligibly). But if you meant there needed to be some set of axioms to get thing s rolling, then I would agree.

↪andrewk

We can draw a parallel between Hume and Godel.

In the early 20th century mathematicians, led by Hilbert, were engaged in a program of proving the soundness of mathematics. Godel proved that that was impossible. Did that mean that Godel claimed we shouldn't use mathematics? Of course not! He thought we should, but just that we should not waste our time trying to prove its foundations were sound.

Ehh, that's not it. Early 20th century mathematicians weren't trying to prove the soundness of mathematics, they were trying to prove its completeness and consistency of it. But as it turned out, you could only have incompleteness or inconsistency. I don't think the analogy holds since the Incompleteness of formalisms capable of expressing number theory doesn't make mathematics rationally unjustifiable. It just means you have to accept that, unless you go with Paraconsistent Mathematics, your mathematical enterprise will be incomplete.

Ying

That is a very contentious proposition, and in any case, I don't see how it bears on warrant. No one denies that we do think - and behave - inductively (except maybe Popperians). — SophistiCat

Umm, no. Popperians wouldn't claim so either:

Block of text

"I had become interested in the problem of induction in 1923. Although this problem is very closely connected with the problem of demarcation, I did not fully appreciate the connection for about five years.

I approached the problem of induction through Hume. Hume, I felt, was perfectly right in pointing out that induction cannot be logically justified. He held that there can be no valid logical arguments allowing us to establish 'that those instances, of which we have had no experience, resemble those, of which we have had experience'. Consequently 'even after the observation of the frequent or constant conjunction of objects, we have no reason to draw any inference concerning any object beyond those of which we have had experience'. For 'shou'd it be said that we have experience' --experience teaching us that objects constantly conjoined with certain other objects continue to be so conjoined--then, Hume says, 'I wou'd renew my question, why from this experience we form any conclusion beyond those past instances, of which we have had experience'. In other words, an attempt to justify the practice of induction by an appeal to experience must lead to an infinite regress. As a result we can say that theories can never be inferred from observation statements, or rationally justified by them.

I found Hume's refutation of inductive inference clear and conclusive. But I felt completely dissatisfied with his psychological explanation of induction in terms of custom or habit.

It has often been noticed that this explanation of Hume's is philosophically not very satisfactory. It is, however, without doubt intended as a psychological rather than a philosophical theory; for it tries to give a causal explanation of a psychological fact--the fact that we believe in laws, in statements asserting regularities or constantly conjoined kinds of events--by asserting that this fact is due to (i.e. constantly conjoined with) custom or habit. But even this reformulation of Hume's theory is still unsatisfactory; for what I have just called a 'psychological fact' may itself be described as a custom or habit -- the custom or habit of believing in laws or regularities; and it is neither very surprising nor very enlightening to hear that such a custom or habit must be explained as due to, or conjoined with, a custom or habit (even though a different one). Only when we remember that the words 'custom' and 'habit' are used by Hume, as they are in ordinary language, not merely to describe regular behaviour, but rather to theorize about its origin (ascribed to frequent repetition), can we reformulate his psychological theory in a more satisfactory way. We can then say that, like other habits, our habit of believing in laws is the product of frequent repetition--of the repeated observation that things of a certain kind are constantly conjoined with things of another kind.This genetico-psychological theory is, as indicated, incorporated in ordinary language, and it is therefore hardly as revolutionary as Hume thought. It is no doubt an extremely popular psychological theory--part of 'common sense', one might say. But in spite of my love of both common sense and Hume, I felt convinced that this psychological theory was mistaken; and that it was in fact refutable on purely logical grounds. Hume's psychology, which is the popular psychology, was mistaken, I felt, about at least three different things: (a) the typical result of repetition; (b) the genesis of habits; and especially (c) the character of those experiences or modes of behaviour which may be described as 'believing in a law' or 'expecting a law-like succession of events'.

A. The typical result of repetition--say, of repeating a difficult passage on the piano--is that movements which at first needed attention are in the end executed without attention. We might say that the process becomes radically abbreviated, and ceases to be conscious: it becomes 'physiological'. Such a process, far from creating a conscious expectation of law-like succession, or a belief in a law, may on the contrary begin with a conscious belief and destroy it by making it superfluous. In learning to ride a bicycle we may start with the belief that we can avoid falling if we steer in the direction in which we threaten to fall, and this belief may be useful for guiding our movements. After sufficient practice we may forget the rule; in any case, we do not need it any longer. On the other hand, even if it is true that repetition may create unconscious expectations, these become conscious only if something goes wrong (we may not have heard the clock tick, but we may hear that it has stopped).
B. Habits or customs do not, as a rule, originate in repetition. Even the habit of walking, or of speaking, or of feeding at certain hours, begins before repetition can play any part whatever. We may say, if we like, that they deserve to be called 'habits' or 'customs' only after repetition has played its typical part; but we must not say that the practices in question originated as the result of many repetitions.
C. Belief in a law is not quite the same thing as behaviour which betrays an expectation of a law-like succession of events; but these two are sufficiently closely connected to be treated together. They may, perhaps, in exceptional cases, result from a mere repetition of sense impressions (as in the case of the stopping clock). I was prepared to concede this, but I contended that normally, and in most cases of any interest, they cannot be so explained. As Hume admits, even a single striking observation may be sufficient to create a belief or an expectation--a fact which he tries to explain as due to an inductive habit, formed as the result of a vast number of long repetitive sequences which had been experienced at an earlier period of life. But this, I contended, was merely his attempt to explain away unfavourable facts which threatened his theory; an unsuccessful attempt, since these unfavourable facts could be observed in very young animals and babies-- as early, indeed, as we like. 'A lighted cigarette was held near the noses of the young puppies', reports F. Bäge. 'They sniffed at it once, turned tail, and nothing would induce them to come back to the source of the smell and to sniff again. A few days later, they reacted to the mere sight of a cigarette or even of a rolled piece of white paper, by bounding away, and sneezing.' If we try to explain cases like this by postulating a vast number of long repetitive sequences at a still earlier age we are not only romancing, but forgetting that in the clever puppies' short lives there must be room not only for repetition but also for a great deal of novelty, and consequently of non-repetition.

But it is not only that certain empirical facts do not support Hume; there are decisive arguments of a purely logical nature against his psychological theory.

The central idea of Hume's theory is that of repetition, based upon similarity (or 'resemblance'). This idea is used in a very uncritical way. We are led to think of the water-drop that hollows the stone: of sequences of unquestionably like events slowly forcing themselves upon us, as does the tick of the clock. But we ought to realize that in a psychological theory such as Hume's, only repetition-for-us, based upon similarity-for-us, can be allowed to have any effect upon us. We must respond to situations as if they were equivalent; take them as similar; interpret them as repetitions. The clever puppies, we may assume, showed by their response, their way of acting or of reacting, that they recognized or interpreted the second situation as a repetition of the first: that they expected its main element, the objectionable smell, to be present. The situation was a repetition-for-them because they responded to it by anticipating its similarity to the previous one.

This apparently psychological criticism has a purely logical basis which may be summed up in the following simple argument. (It happens to be the one from which I originally started my criticism.) The kind of repetition envisaged by Hume can never be perfect; the cases he has in mind cannot be cases of perfect sameness; they can only be cases of similarity. Thus they are repetitions only from a certain point of view. (What has the effect upon me of a repetition may not have this effect upon a spider.) But this means that, for logical reasons, there must always be a point of view--such as a system of expectations, anticipations, assumptions, or interests--before there can be any repetition; which point of view, consequently, cannot be merely the result of repetition. (See now also appendix *X, (1), to my L.Sc.D.)

We must thus replace, for the purposes of a psychological theory of the origin of our beliefs, the naïve idea of events which are similar by the idea of events to which we react by interpreting them as being similar. But if this is so (and I can see no escape from it) then Hume's psychological theory of induction leads to an infinite regress, precisely analogous to that other infinite regress which was discovered by Hume himself, and used by him to explode the logical theory of induction. For what do we wish to explain? In the example of the puppies we wish to explain behaviour which may be described as recognizing or interpreting a situation as a repetition of another. Clearly, we cannot hope to explain this by an appeal to earlier repetitions, once we realize that the earlier repetitions must also have been repetitions-for-them, so that precisely the same problem arises again: that of recognizing or interpreting a situation as a repetition of another.

To put it more concisely, similarity-for-us is the product of a response involving interpretations (which may be inadequate) and anticipations or expectations (which may never be fulfilled). It is therefore impossible to explain anticipations, or expectations, as resulting from many repetitions, as suggested by Hume. For even the first repetitionfor-us must be based upon similarity-for-us, and therefore upon expectations--precisely the kind of thing we wished to explain.

This shows that there is an infinite regress involved in Hume's psychological theory.

"Hume, I felt, had never accepted the full force of his own logical analysis. Having refuted the logical idea of induction he was faced with the following problem: how do we actually obtain our knowledge, as a matter of psychological fact, if induction is a procedure which is logically invalid and rationally unjustifiable? There are two possible answers: (1) We obtain our knowledge by a non-inductive procedure. This answer would have allowed Hume to retain a form of rationalism. (2) We obtain our knowledge by repetition and induction, and therefore by a logically invalid and rationally unjustifiable procedure, so that all apparent knowledge is merely a kind of belief--belief based on habit. This answer would imply that even scientific knowledge is irrational, so that rationalism is absurd, and must be given up. (I shall not discuss here the age-old attempts, now again fashionable, to get out of the difficulty by asserting that though induction is of course logically invalid if we mean by 'logic' the same as 'deductive logic', it is not irrational by its own standards, as may be seen from the fact that every reasonable man applies it as a matter of fact: it was Hume's great achievement to break this uncritical identification of the question of fact--quid facti--and the question of justification or validity--quid juris. (See below, point (13) of the appendix to the present chapter.)

It seems that Hume never seriously considered the first alternative. Having cast out the logical theory of induction by repetition he struck a bargain with common sense, meekly allowing the reentry of induction by repetition, in the guise of a psychological theory. I proposed to turn the tables upon this theory of Hume's. Instead of explaining our propensity to expect regularities as the result of repetition, I proposed to explain repetition-for-us as the result of our propensity to expect regularities and to search for them.

Thus I was led by purely logical considerations to replace the psychological theory of induction by the following view. Without waiting, passively, for repetitions to impress or impose regularities upon us, we actively try to impose regularities upon the world. We try to discover similarities in it, and to interpret it in terms of laws invented by us. Without waiting for premises we jump to conclusions. These may have to be discarded later, should observation show that they are wrong.

This was a theory of trial and error--of conjectures and refutations. It made it possible to understand why our attempts to force interpretations upon the world were logically prior to the observation of similarities. Since there were logical reasons behind this procedure, I thought that it would apply in the field of science also; that scientific theories were not the digest of observations, but that they were inventions--conjectures boldly put forward for trial, to be eliminated if they clashed with observations; with observations which were rarely accidental but as a rule undertaken with the definite intention of testing a theory by obtaining, if possible, a decisive refutation."
-Karl Popper, "Conjectures and Refutations", p. 55-61

andrewk

↪MindForged

Early 20th century mathematicians weren't trying to prove the soundness of mathematics, they were trying to prove its completeness and consistency of it. But as it turned out, you could only have incompleteness or inconsistency

I know. I just didn't want to use technical terms like completeness and consistency in a discussion that has not been heavily technical thus far.

andrewk

↪Janus

I would not agree with classing Hume as a pragmatist. Perhaps you just meant that he was a pragmatic thinker?

Yes that's what I mean, which is why I carefully avoided using a capital P that would imply similarity to Peirce, James and Dewey. I happen to think there are some similarities but it doesn't matter to this discussion whether there are, or how deep they go, and I think it would be a distraction to get into that.

MindForged

↪andrewk

Ah, my mistake then.

ssu

Ehh, that's not it. Early 20th century mathematicians weren't trying to prove the soundness of mathematics, they were trying to prove its completeness and consistency of it. But as it turned out, you could only have incompleteness or inconsistency. I don't think the analogy holds since the Incompleteness of formalisms capable of expressing number theory doesn't make mathematics rationally unjustifiable. It just means you have to accept that, unless you go with Paraconsistent Mathematics, your mathematical enterprise will be incomplete. — MindForged

Gödel shows how limited is our ability to give direct proofs. (Just like, well, Turing did also.) Gödel's theorems simply show how tricky self-reference (which with Gödel doesn't end up in a Paradox) is and thus the idea of there being a way to prove everything that is true to be so is simply false. That doesn't at all make Mathematics unlogical.

Some sciences do have the problem of self reference or subjectivity. Just think about the social sciences: if the findings in social sciences like economics or sociology themselves have an effect on how we behave, how we understand ourselves and how we make decisions, then it has a "problem" of self reference. This is totally evident in things like economics where there allways is a normative side to the subject along with the descriptive.

Yet this doesn't at all mean that the study of societies wouldn't be a scientific endeavour.

SophistiCat

Sorry, i meant warrant being epistemologically vacuous. — sime

I still have no idea what you mean by this. Warrant is what makes epistemology normative. To say that such and such belief is warranted is to say that you can and should believe such and such. What is vacuous about this?

sime

But Hume represents the nominalist turn of thought. He was not a pragmatist in the sense of arguing for the reality of the general or universal. He was an atomist in regards to empirical sense data. So his epistemology reflects a particular brand of metaphysics. — apokrisis

Whether or not Hume was an atomist is irrelevant, as Goodman's new riddle of induction illustrates.

If today one person sees an object as green, and another person sees it as grue (i.e. currently green up until some future time t, then blue afterwards), then their principles of the uniformity of nature are different.

As this illustrates, the so called 'principle' of the uniformity of nature is relative to one's ontology, and hence so is one's principle of induction. And regardless of whatever this ontology is, the infallibility of induction relative to this ontology cannot be non-circularly justified, nor empirically defended.

In my opinion a better way to understand Hume, is to say that whatever one uses as a principle of induction it is impossible to distinguish good from bad inductions without pain of circularity.

sime

I still have no idea what you mean by this. Warrant is what makes epistemology normative. To say that such and such belief is warranted is to say that you can and should believe such and such. What is vacuous about this? — SophistiCat

I'm saying that if there are no objective criteria, i.e. physical criteria, for ascribing to agents propositional-attitudes pertaining to prediction-making, then it makes no objective sense to discuss agents as needing epistemological warranty for induction, since applications of rules of induction is then in the eye of the beholder, for example the community the agent belongs to who selectively interprets his behaviour as prediction-making for their own concerns.

In other words, I am suggesting that to follow a rule of induction is no different to following any other rule; it is a normative principle pertaining to language-games, but not in any way that is significant to metaphysics or epistemology.

MindForged

↪ssu

Gödel shows how limited is our ability to give direct proofs. (Just like, well, Turing did also.) Gödel's theorems simply show how tricky self-reference (which with Gödel doesn't end up in a Paradox) is and thus the idea of there being a way to prove everything that is true to be so is simply false. That doesn't at all make Mathematics unlogical.

...yea? I didn't say Godel's results made math "unlogical", I said his Incompleteness theorems entail that any sufficiently expressive formal system (i.e. one capable of arithmetic) must be either incomplete or inconsistent. In other words, there's a limitation of what sorts of desirable properties such an enterprise can have. Paraconsistent Mathematics allows one to (non-trivially) maintain Completeness, but it's inconsistent (this is too far for some people). Standard mathematics retains consistency (well, no known inconsistencies anyway), and as such is necessarily incomplete. That's all I said, so I don't think we disagree.

aletheist

objective criteria, i.e. physical criteria — sime

What exactly do you mean by "physical criteria"? What is your warrant for equating objective criteria with physical criteria?

ssu

...yea? I didn't say Godel's results made math "unlogical", I said his Incompleteness theorems entail that any sufficiently expressive formal system (i.e. one capable of arithmetic) must be either incomplete or inconsistent. In other words, there's a limitation of what sorts of desirable properties such an enterprise can have. Paraconsistent Mathematics allows one to (non-trivially) maintain Completeness, but it's inconsistent (this is too far for some people). Standard mathematics retains consistency (well, no known inconsistencies anyway), and as such is necessarily incomplete. That's all I said, so I don't think we disagree. — MindForged

No, we don't.

As typical with Mathematics, you can say something differently from just another point of view and still both are correct.

MindForged

↪ssu

Well I missed where the disagreement was then, we both said that Godel showed a limitation on the ability to give proofs in mathematics. Sooo, eh, whatever.

Akanthinos

Couldn't we just abandon the idea of internal consistence, like Tarski did, and then keep on doing logic in whatever other meta-language provides external consistency to maths? :-|

SophistiCat

No one denies that we do think - and behave - inductively (except maybe Popperians). — SophistiCat

Umm, no. Popperians wouldn't claim so either: — Ying

I am not sure why you posted this lengthy excerpt in response to my off-hand remark. In it Popper criticizes Hume's psychological account of induction, ending up endorsing the view that "We obtain our knowledge by a non-inductive procedure."

First of all, I wasn't referring to Hume specifically. By "inductive thinking" I meant our tendency to identify patterns of occurrences and extrapolate them beyond the available concrete facts, both as a way of explaining what we know and of predicting what we don't know.

You might instead argue that Popper rather overstates his opposition to induction, and that his own view is distinguished only in some particulars from the common-sense take that I outlined above. But this is not all too obvious from your quote.

Anyway, this is all very dated stuff, of interest mainly to historians.

SophistiCat

In other words, I am suggesting that to follow a rule of induction is no different to following any other rule; it is a normative principle pertaining to language-games, but not in any way that is significant to metaphysics or epistemology. — sime

Of course induction is normative - I don't understand why you keep saying this as if this is something controversial. And your last remark makes me wonder what you think metaphysics and epistemology are about.

Janus

That axioms are not proveable does not entail that they are not rationally warranted. They are rationally warranted because without them there can be no discourse. The irrational demand for absolute proof is the whole source of these kinds of humean errors of thought.

I think I was mostly with you until you said this (depending on what you meant). If by this you meant a particular set of axioms are rationally warranted because without them discourse is impossible, I would find that dubious (people disagree about what axioms should be adopted in math and logic, and they do so intelligibly). But if you meant there needed to be some set of axioms to get thing s rolling, then I would agree. — MindForged

I was not referring to any "particular set of axioms" as being indispensable, although it is arguable that there are some axioms that seem to be fundamental to human experience; and that consequently seem self-evident, and anyone can intuitively 'get' them. The axioms of Euclidean geometry would seem to fall into this category. Of course, non-Euclidean geometries exist, but they are not intuitive in the 'direct' way that Euclidean geometry is.

MindForged

↪Janus

I was not referring to any "particular set of axioms" as being indispensable, although it is arguable that there are some axioms that seem to be fundamental to human experience; and that consequently seem self-evident, and anyone can intuitively 'get' them. The axioms of Euclidean geometry would seem to fall into this category. Of course, non-Euclidean geometries exist, but they are not intuitive in the 'direct' way that Euclidean geometry is.

But doesn't that show the weakness of this view (relying on intuition to settle the matter)? Intuition doesn't really seem to lend much justification to accept what we are "getting" via it, and worse, intuition can cause us to dismiss truths which conflict with them. You mention Euclidean Geometry as being direct and intuitive, but some of our best scientific theories suggest we have to understand the space of the universe as being Non-Euclidean (Relativity comes to mind).

That's not to say intuitions aren't important, but I'd probably not hold them in too high a regard when discussing "deeper" issues.

Janus

↪MindForged

I agree it is arguable that non-intuitive scientific models and theories may show us a "deeper" structure of reality (whatever we might think that means) but when it comes to rational justification, in the sense of the logical structures, of our everyday discourse about the world the foundational axioms that make that discourse possible are mostly intuitively given, I would say.

Science is just a re-branding of logic

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