Can Hume's famous Induction Problem also be applied to Logic & Math?
I really do not get it, why Hume judged at his time, it was possible at any time, the sun could not rise tomorrow (experience as an unreliable source of knowledge), but not equally, that from p and p -> q from tomorrow no more q follows (logic as an unreliable source of knowledge). What prevents us from imagining that we all wake up tomorrow and apply other logical rules? What prevents us from imagining that we all wake up tomorrow and a circle is no longer round, because we find ourselves in a chaotic (inconsistent) world in which the definiendum no longer entails the definiens and vice versa?
Comments (58)
Perhaps the first thing to point out is that it's not at all clear what this could possibly mean: if a definiendum no longer entails its definiens, in what sense could the one count as a definiendum, and the other, its definiens? What could it mean to say that tomorrow, bachelors might no longer be unmarried men? Does it mean that they will all be married (by forced decree, perhaps?): but this would be an 'empirical' change. So it can't be that. But if not that, then what? But that's what Hume was concerned with: experience. But if not experience, then - it's not clear what it could even mean to extend the problem of induction to logic and math.
However when considered ontologically, logical deduction is also called into question. The underlying issue is that there doesn't exist agreement regarding the relationship of deduction to induction. For example, deduction might be considered to be special case of induction in which there is believed to exist perfect certainty for a conclusion with respect to a given premise.
Some philosophers identify deduction with the semantic notion of synonymy, yet for most people for most of the time, deduction is identified with the consequences of physical calculation as demonstrated by our reliance on computers. Hence deduction in practice is treated as a special case of (fallible) induction.
That is the way the world is. There is no certainty. Certainty about the real world is an illusion or perhaps a delusion.
What are these destructive conclusions of which you speak?
Knowing that we'll not call squares by any other name...
If I understand correctly, you are asking if, if it is possible for some constant to change, then can the laws of logic change too?
The answer is no, because asking if a thing is "possible" is to ask if it is "logically possible". Thus the statement "it is logically possible for logic to change" is nonsense. The laws of logic is the reference point around which everything can possibly change. The reference point cannot change around itself.
Nowadays, Hume's intuition about the sun is considered to be quite right:
The Solar System will remain roughly as we know it today until the hydrogen in the core of the Sun has been entirely converted to helium, which will occur roughly 5 billion years from now. This will mark the end of the Sun's main-sequence life. At this time, the core of the Sun will contract with hydrogen fusion occurring along a shell surrounding the inert helium, and the energy output will be much greater than at present. The outer layers of the Sun will expand to roughly 260 times its current diameter, and the Sun will become a red giant. Because of its vastly increased surface area, the surface of the Sun will be considerably cooler (2,600 K at its coolest) than it is on the main sequence.[51] The expanding Sun is expected to vaporize Mercury and render Earth uninhabitable.
Quoting Pippen
As long as P does not change, then Q will keep necessarily following. If P is a Platonic abstraction, then there are no reasons for it to change.
For example, Pythagoras' theorem will forever remain provable from Euclid's classical axioms. You would have to modify Euclid's Platonic abstractions, i.e. the basic beliefs (axioms) that construct the abstract world of classical geometry, to effect a change that would render Pythagoras' theorem unsound.
You could also try to modify the axioms of first-order logic to make the inferences, in the proof for Pythagoras' theorem, invalid.
The difference between Hume's physical world and the abstract, Platonic worlds on which logic operates, is that there are no changes possible outside our control in abstract, Platonic worlds.
Quoting Pippen
You actually can. Hilbert calculi are an exercise on doing exactly that:
In logic, especially mathematical logic, a Hilbert system, sometimes called Hilbert calculus, Hilbert-style deductive system or Hilbert–Ackermann system, is a type of system of formal deduction attributed to Gottlob Frege[1] and David Hilbert. These deductive systems are most often studied for first-order logic, but are of interest for other logics as well. Most variants of Hilbert systems take a characteristic tack in the way they balance a trade-off between logical axioms and rules of inference.[1] Hilbert systems can be characterised by the choice of a large number of schemes of logical axioms and a small set of rules of inference.
It does not seem to be possible to create more powerful systems of logic by adding axioms:
Because Hilbert-style systems have very few deduction rules, it is common to prove metatheorems that show that additional deduction rules add no deductive power, in the sense that a deduction using the new deduction rules can be converted into a deduction using only the original deduction rules.
In fact, this is a well-known phenomenon in mathematics. Beyond the basic construction, adding axioms usually does not make a system stronger. The system rarely knows more through these new axioms. It will just trust more.
For example, arithmetic theory can "see" all theorems and proofs of all other systems, including the "more powerful" ones, such as set theory. So, it actually knows these theorems, but it does not trust them. More axioms often means that your system does not become more knowledgeable or more powerful, but only more gullible.
I just said, didn't I? (Assuming you are replying to me.)
Said what?
No, that's not how problems usually work.
Quoting A Seagull
The undermining of knowledge. Biting the bullet is admitting that the ancient skeptics were right.
Which gives logic and math a kind of atemporal, aspatial quality. Which is odd, given that we inhabit temporal, spatial universe of change.
This makes sense only if you ignore the difference between probability and certainty. Any conclusion with a probability less than a 100% is from an inductive argument and anything else (100% probabilty/certainty) is from a deductive argument.
Also induction has fewer forms among which arguments from analogy and statistical arguments are the only ones I remember.
Deduction, on the other hand, probably has an infinite number of valid forms. If not they at least outnumber the forms available in induction.
If I recall correctly the problem of induction, although exposing a limitation of inductive [i]logic[/I], is more about a specific class of induction viz. science where statistics plays a huge role as conclusions about how nature works are drawn from a finite sample space of observations.
1. The present will resemble the past.
Why?
Because
2. The present has resembled the past
Why?
Because
3. The present has resembled the past
As you can see this is a circular, ergo fallacious, deductive argument.
https://thephilosophyforum.com/discussion/comment/309821
Why? P, P -> Q | Q is just right because it follow from some rules. But these rules can change overnight, can they? So MP could be true today but false tomorrow. Imagine - overnight - our world becomes weird in the way that it becomes impossible to construct any implication P -> Q (~P v Q). I know it's hard to imagine, but I can just write it down and say: so be it from henceforward. Obviously in such a world you could not conclude anymore Q from P & P -> Q because it wouldn't be a wff at all. But somehow Hume and basically all philosophers after him disregard such a possibility.
It doesn't follow from a rule in the sense of a social construct. Something that people decided to do. It follows from the way human minds work. It's possible human minds change overnight, but we, being human minds, wouldn't notice.
Quoting Pippen
The world cannot change logic, other than changing human minds. You can write it down, but can you actually believe it?
Propositional logic is an axiomatic theory, derivable from 14 arbitrary, speculative beliefs with no justification. We have no clue as to why these axioms are the starting-point beliefs of this system. If we knew, then they would not be axioms.
In the Platonic view, we sense that these axioms may have some connection somehow to the physical universe. In the intuitionistic view, we argue that these axioms are somehow connected to our natural predisposition to believe them.
In every case, however, we commit to refrain from trying to justify axioms from within mathematics, unless we can replace them by more fundamental axioms. That would, however, still not change the axiomatic nature of the system.
Since we do not know why these axioms are there in the first place, we also do not know on what grounds they could change. Hence, your question is fundamentally undecidable.
Again, I don't understand why the induction problem is never seen as problem for logic and math as well, there must be something I do miss since far wiser people than me dealt with this before.
Hume clearly differentiated between what he called 'matters of fact', ie facts about the real world, and what he called 'relations of ideas' ie abstract logical systems such as mathematics.
This differentiation separates induction from deduction, the real from the abstract.
Your ‘i.e’ is yours alone, Hume never used such an expression. He distinguished analytical and empirical but never used the expression ‘the real world’.
There is a reason why necessary truths are held to be true ‘in all possible worlds’. specifically, that no world in which such truths did not obtain could not exist. So your supposition is mere idle wordplay.
My takeaway is rather different: not that math and logic are atemporal and aspatial, but rather, that they are normative practises, techniques, employed and tailored for certain purposes, outside of which certain questions simply no longer make sense. Consider the OPs example: the idea that a definiendum no longer entails its definiens. If someone were to say this, the only possible response to make is that he or she does not understand what either or definiendum or a definiens is. That he or she does not understand how language is used in these cases.
Or else: if a 'circle is no longer round' - what could this mean? Either, on the side of 'language', that people no longer call what they used to call circles, circles (they call them 'kirkles' now). Or, on the side of 'things', that everything circular has changed shape (to decagons, say). But these two options exhaust the space of manoeuvre: circles just are round - not because of some deep, metaphysical necessity, but because that is how we use language. If someone says: 'but circles might not be round tomorrow' - the only response is: 'you don't understand what circles are'.
Nothing prevents us from imagining that.
Quoting Pippen
The problem of induction is about the justification of laws. It's a problem for knowledge internalists who insist that in order to know something (a law, for instance), one would have to have access to at least part of a justification.
I'm not a mathematician, so I don't know to what extent math is made of laws the way physics is. I'm not sure the method for determining the circumference of a circle is thought of as a law. But in imagining that circles might not be round tomorrow, the issue isn't math anyway. At first glance, it sounds like we're imagining that language use changed.
There's nothing mysterious here. Logic and math are our constructs, and as ideas that we entertain our minds, write down and communicate to each other, they originate in time, change and disappear - same as with our ideas about the world. But there is a fundamental disanalogy here between purely abstract ideas and ideas about the world: abstract ideas are not about anything, so they don't answer to the challenge of observation, which is what the problem of induction is all about.
ETA Or what said.
I'm not sure what you mean by "abstract ideas." An abstract object is by definition a thing which various individuals note and can be wrong about. Mathematical objects are abstract objects. They do exist.
The so called 'laws of physics' are not really 'laws' in that physical entities are compelled to follow those laws. What you actually have in the 'laws of physics' are a description, and often a very accurate description, of the way that physical entities behave.
The only 'justification' required for a distinction is that it proves useful for the understanding of the world.. such as the difference between mammals and fish so that dolphins belong to one but not the other.
As for Hume's distinctions.. I was referring to his famous quote regarding 'matters of fact' and 'relations of ideas' and that if a book in a library said nothing about either it should be burnt.
I am just extrapolating, or perhaps bringing up to date this distinction, so that it differentiates between ideas or statements that refer to the real world or those that refer to an abstract world.
All too often in philosophy people conflate the two and arrive at false conclusions about the real world.
We can't describe events in the future because we dont have access to them. Laws of physics are expected to predict events, so in what sense could we say they're descriptions?
That's all the average person thinks of as a physical law: predictability, the compulsion angle is funny.
We can say that the laws of physics can be used to predict a description of the future.
If you lived in a world where nothing followed necessarily from anything else, then MP wouldn't apply. Perhaps, for example, a quantum vacuum where particles just pop into and out of existence and you are a Boltzmann brain.
However that wouldn't be a case of MP failing, merely that it has no application or use in that scenario. Whereas the rule p; p ->q; not q is inconsistent and so couldn't apply to any scenario.
Quoting alcontali
The sun will still be around in five billion years, therefore Hume was right that the sun might not rise tomorrow? :-)
Technically, yes. Some day it will be true.
So we dont predict future events, we predict descriptions of future events?
:chin:
No, we would just look for ways to model the world that avoided inconsistency. Which is just what occurs in quantum mechanics where the state of a particle in superposition is represented as the sum of two or more distinct states.
What that means is a separate question.
But your answer implies that it could happen and so we'd need to adjust our logic and math, but that means the problem of induction also applies to logic and math so why did Hume not agree? For him only our experiences are inductive, not logic and math, but we see that it's wrong. We can easily imagine a scenario where our world of tomorrow has different logical/math rules than the world of today. Was David that blind?
As I see it, if the world seemed inconsistent then that would point to a problem of representation, not that the world was inconsistent. That's the approach physicists have taken with quantum mechanics since it is standardly represented using classical logic and algebra (though, of course, new mathematics such as matrix mechanics was developed).
Quoting Pippen
As you may know, Hume drew a distinction between "relations of ideas" and "matters of fact and real existence". For the broad ramifications of that, see Hume's fork. If one inquires whether the "relations of ideas" are themselves "matters of fact" then the distinction starts to break down or become circular. So the problem of induction only makes sense in a context of deductive certainty per Hume's distinction.
If that distinction is rejected, then you are left with something like an Aristotelian view (or pragmatic view) where the meaning, or use, of terms like "possible" are understood in the context of the law of non-contradiction or ostensive demonstration. In turn, accepting inconsistency seems to take one out of the bounds of meaningful language. But one needs meaningful language to state one's position in the first place.
Not really. It just means trivialism, i.e. everything becomes true, but I can still talk and be understood on an ostensive level. That sounds enough meaningful to me.
Back to Hume. How does he prove that logic is not a matter of fact but something higher? IMO he can't, so his "fork" is pretty much made up from speculation and tradition.
On trivialism, the statement of mine that you are disagreeing with is trivially true. So your disagreement assumes non-trivialism (i.e., you think my statement is false). Can you give an example of ostensive talk that doesn't assume non-trivialism?
Quoting Pippen
I think you're right.
I can't, you are right.
My line of thinking goes like this: Imagine tomorrow the proposition "p & ~p" becomes somehow true! At that point our logic would collapse, we couldn't even talk or think about it (trivialism), we'd be literally insane. But we can today - under the assumption that our logic still holds - verify that this very scenario could happen tomorrow and that it would destroy our logic. Isn't that a proof by example that logic is not necessary?
A possible analogy: The rules of chess specify that players can only move a bishop along the diagonal. Moving the bishop vertically would be invalid. Suppose, tomorrow, a player moves the bishop vertically. One response is to reject the move. Another response is to change the rules of chess to accommodate the move.
The conventional rules of logic include the law of non-contradiction (LNC) which specifies that it is impossible for states-of-affairs p and ~p to obtain simultaneously. Or, propositionally, that propositions p and ~p can't be true simultaneously.
Suppose per hypothesis that, tomorrow, "p & ~p" becomes true. One response is to reject the hypothesis. That is, to say that such a scenario is impossible and thus cannot obtain. Another response is to change the rules of logic to accommodate the scenario.
Now the "changing the rules" response - rejecting the LNC - collapses to trivialism. (Note: unless explosion is also rejected, as with paraconsistent logics.)
Whereas under the existing rules the hypothesis that, tomorrow, "p & ~p" becomes true is, per the LNC, provably false. Thus, according to the existing rules, this scenario couldn't happen tomorrow.
Quoting Pippen
Not if the rules of logic include the LNC. But note, as with the chess example, one can always propose one's own rules. However they may have undesirable consequences (such as trivialism).
How do we know there is a problem with induction when the problem itself is empirical and based on experience? You're using induction to prove the problem of induction exists.
The problem of induction isn't much different than the paradox of knowing that you know nothing. We need a better definition of knowledge to solve the problem of induction and the paradox.
Both impossible if p & ~p becomes true because then we couldn't talk/think straight/meaningfully.
More and more it seems that the main difference between induction and deduction is this: If we imagine an induction to be false tomorrow then we can at least comprehend what it would mean for tomorrow while with deduction we could only say that if it happens then tomorrow would 'black out' for us. But that's not enough for me to exclude such a possibility. It would be like if you would exclude the possibility to get insane tomorrow just because you couldn't say anything about that tomorrow.
The same thing that prevents you, or should prevent you, to imagine our planet is actually called Penis, while knowing for a fact it is named Earth. Nonsense.
Interesting theory. How does it deal with the fact of me (or some Humpty) claiming to call it whatever I like?
Sure. My difficulty was with making sense of,
Quoting Zelebg
How to know such a fact. Perhaps you meant, agreeing to assume?
Something like that, but calling circle “round” is not about making assumptions, it is about putting a label on a specific and certain geometrical relation.
Nonsense in imagining that circle is no longer round is due to using two terms from one formal system which are linked together and then moving one of them to the context of some new undefined reference chart of meanings. This is just an exercise in messing with the English dictionary and produces gibberish. Imagine a flower is a vacuum cleaner.
Ok. I did that.
It makes sense in some abstract fantasy context, as a painting for example. But circles that are not round you can not even paint because the two terms are semantically linked so strongly they are almost synonyms. It’s like trying to paint a flower that is not a flower. It's gibberish, a contradiction, it does not compute, and being semantically invalid statement it can not be sanely reasoned about.
So, on waking that morning (OP), we might all seize and catch fire like confused robots.
But we might do that anyway if we took any logic too religiously, and felt obliged to believe all the consequences of our (inevitably) inconsistent beliefs.