Hacker News Reader

>What then is a mathematical truth

In the West, Aristotle put it as follows: "To say of what is that it is not, or of what is not that it is, is false, while to say of what is that it is, and of what is not that it is not, is true"

Some 2 millennia later, Tarski, a Polish logician, more or less formalised just that. This result, among other things, spawned a brand new branch in mathematics: Model Theory. More on Tarski’s notion of truth: https://plato.stanford.edu/entries/tarski-truth/. A few decades later some analytic philosophers like Davidson drew on Tarski’s work and applied it to natural language, too. This resulted in this: https://en.wikipedia.org/wiki/Truth-conditional_semantics. But notwithstanding Tarski’s work, there’s still significant controversy surrounding both “truth in natural language” and the logico-mathematical truth predicate in philosophical circles. Here’s an article that covers axiomatic theories of truth in general: https://plato.stanford.edu/entries/truth-axiomatic/.

The clock problem is, as it seems most philosophical problems I seem to encounter, one of definitions. It's a game with words. It doesn't change the underlying facts. You had a good reason to believe proposition X, and X happened to be true, but your reason to believe X was not causally linked to X's fact-ness. Whether you want to call that circumstance "knowledge" or use some other symbol to describe it does not change the underlying fundamentals at all. And so all the millions of hours of thought that have been spent on this problem have been wasted.

This mistake of reifying of non-fundamental concepts and then thinking about them without reference to their component parts is an epidemic in philosophy, and it always has been.

Imagine you are designing a software system. One use case requires you to model a fact as best as you can, so you write a fact-class for it. Your software system runs fine, but you notice the class doesn't accurately model the fact in some other use cases. You now know the class is insufficiently accurate. Do you still trust the result of your software?

Replace software with language, class with concept we use to refer to facts and you have to claim that any further thought into the design of your class is wasted, because you hold that to be the case for concepts, language and reality.

If all philosophers are talking about is how underspecified words are, that is fine, but there's little value in it. Language is not designed. So a word is underspecified. Nobody can fix it.

But I was under the impression that philosophers thought themselves more than mere dictionary debaters. Whenever I hear about this problem, my impression is that at least non-zero of them think there is a problem with the actual concept of knowledge, not merely the word.

we're just into another 'what's truth' cycle. next will be 'is truth communicable' then 'is truth subjective'

The problem with this dismissal is that "knowledge" is not just a word, it does in fact refer to something solid and concrete (insofar as most of us intuitively believe there is a difference between genuine knowledge and just belief).

In the real world it is impossible to find human experiences that are "causally linked" to facts in the world, and so the implication of the argument is that EVERY experience you have ever had is possibly like looking at the broken clock.

And yet, most of us would believe that there is a difference between looking at a broken clock that says noon, versus a working clock that says noon. It would be nice to have a rigorous way to distinguish between the two cases - solving that problem might teach us something about the world.

>The problem with this dismissal is that "knowledge" is not just a word

"Knowledge" is just a word. It references many different concepts.

> In the real world it is impossible to find human experiences that are "causally linked" to facts in the world

Either you've misunderstood what causal linkage is, or else you are some odd brand of solipsist, but either way, this is incorrect. Causal linkage just means that that event participated in causing this event. The sun being in the sky is causally linked to me believing it's day time, for example.

> And yet, most of us would believe that there is a difference between looking at a broken clock that says noon, versus a working clock that says noon.

Of course there is. In one case, your belief is causally linked to fact. In the other case, it isn't, but still happens to be correct.

> Causal linkage just means that that event participated in causing this event. The sun being in the sky is causally linked to me believing it's day time, for example.

Great example! In Ptolemy's time "the sun being in the sky" would have meant that the sun was in a particular phase in its orbit around the Earth. Now we understand that the sun is not actually "in the sky" but it is a hundred million kilometres away. Our understanding of what it means for the sun to be "in the sky" will likely undergo several revolutions in the next 100,000 years.

Second, the actual facts are not causally linked to your belief. Your perception that the sun is in the sky is causally linked to your belief that it is daytime. If I am a schizophrenic and I hallucinate that the sun is in the sky even though it is night time then do I "know" that it is daytime?

All of this probably sounds overly pedantic but teasing apart the details is exactly what we're trying to do here :)

(Same person, different account.)

> Our understanding of what it means for the sun to be "in the sky" will likely undergo several revolutions in the next 100,000 years.

I wouldn’t bet on it.

> Your perception that the sun is in the sky is causally linked to your belief that it is daytime.

My perception is cause by electrical impulses from my eyes, which are caused by photons hitting those receptors. The photons themselves are caused by the sun, and their incoming vector is caused by its position relative to me. There is a causal chain, thus, causal linkage.

> If I am a schizophrenic and I hallucinate that the sun is in the sky even though it is night time then do I "know" that it is daytime?

Depends on what you mean by know. Your perceptions are not causally linked to a true belief, if that is what you are asking.

Gettier cases long ago convinced me that it is useless to talk about 'truth' apart from 'justification'.

Obviously, none of us can ascertain truth directly. The only way we can claim A or B is true is by appealing to some justification for our belief. Therefore, the idea that [knowledge = true and justified belief] collapses to [knowledge = justified belief].

From that point, we have only to discuss what constitutes better/worse justification(s).

I agree with you, but we have to be aware that by this definition two diametrically opposed "knowledges" can be equally valid if they have disjoint justification frameworks.

And that leads to relativism, which we don't want.

Tell that to physicists who have been grappling with the limitations of the classical model for the last 50 years ... we can model our own directly perceived reality well enough but once we get out into the depths (e.g. quantum levels) it all breaks down. My laymans opinion is not that these systems are necessarily unpredictable or unstable, just that they're so far outside the scope of our normal experience we simply can't comprehend them in any intuitive way. Save for a few gifted souls of course!

The classic "Rationalist" (as in Kant) explanation of mathematics is that things like pi are an a priori concept, or phenomena. An ideal if we will, that is innate within us that helps us to explain the reality without (the noumena). Mathematics relates to the interrelation of these concepts. Physics is a (very successful) attempt to relate these apriori phenomena concepts directly to the noumena.

Again, just a have-a-go layman on the philosophical front as well and I welcome any corrections or more sophisticated elaborations!

The word "truth" is loaded. Which is a pity, because the notion is pretty simple. Here's a simple proposition:

There is a world out there. This reality, universe, or whatever, is in some state or another, works in some way or another, and whatever we think about what it is or how it works has little bearing on what it actually is, or how it actually works. "Truth" then is merely an accurate depiction of our world (or parts thereof). The problem is ensuring accuracy, and that is impossible without infinite evidence. We can only approach certainty.

And there is indeed a point where we are certain enough, and further inquiry isn't worth the hassle.

> whatever we think about what it is or how it works has little bearing on what it actually is

We are a part of this world, so if you accept that you must accept that how you think about the world is a part of the definition and state of the world itself.

Which quickly leads to the conclusion that it is impossible to reason about a system from inside the system with absolutely certainty.

> We are a part of this world,

Hence "little", instead of "no".

My problem with the example is I don't understand how this philosophical problem can possibly translate to a language with more refined words for "know" and "knowledge." To me, it appears to be bickering over the definition of an English word. Happy to be shown why this isn't the case.

I believe I can make it a non-problem by defining some new words.

Belief - a thing one thinks is representative of the universe

Knowledge - a thing one thinks is representative of the universe, with justification

Billabooknowledge - a thing one thinks is representative of the universe, with justification, that turns out to be wrong, despite one not being aware it's wrong

I had the same thought: the problem seems to be the English word "know". In ordinary use the word does seem to imply "with justification", but it's unclear what should count as "justification", seeing as ordinary (non-philosophical) thought involves so much intuition and analogy. So perhaps it's a word philosophers should avoid rather than waste time worrying about. But who am I to advise philosophers how to spend their time?

Is knowledge not just the reliable predictability of a model’s outputs based on inputs?

I.e “you know it is noon” means you can predict that when you look outside the window it will be light, the church bell will be ringing, the clock will point to 12 etc. One or more of these corroborating factors provides that knowledge to you (eg you can still know it’s noon if the clock is broken because you can hear the church bells and see the sun above your head etc)

This is exactly what intuitionistic logic is all about. Instead of doing operations on variables which represent the truth value of a proposition, we can do operations on variables which represent the justification for a proposition. All we have to do is forego the law of excluded middle (A or not A is true). Which makes sense because we can't say we always have justification for A or have justification to refute A. Sometimes there are questions you can ask which you can never find the answer for or constrict justification.

> There are only models and predictions, and some models lead to better predictions than other models.

"Better prediction" requires evaluating a truth claim, so this viewpoint would seem to contradict the rejection of truth.

Indeed. I’d say more than models and predictions, what really defines “truth” is consensus in what are and are not acceptable beliefs.

As Mac pointed out in IASIP: Reynolds vs. Reynolds, you need a leap of faith to accept all the sciency writings about subatomic particles and whatnot.

I think you're highlighting the difficulty of capturing precise truths, and conflating theories with truth.

For example:

* As of writing this comment, I have five fingers on each of my hands.

* Humans must stay hydrated to survive.

* Within my office, the time is currently 4:45 +/- 1 minute.

Those are truths, though there may be pedantic holes in my communication.

> As of writing this comment, I have five fingers on each of my hands.

Well, if you are not a brain in a jar, maybe. As far as I am concerned, I don't have any undeniable evidence that you even exist, nor that I have hands.

That's a pretty useless kind of skepticism, but as long as you are going for objective "truth" you must face that kind of problem.

you only “must” face that “problem” when dealing with people who are either overintellectualizing uselessly, as you note, or with assholes. the distinction is slim!

Even then it depends on your definition of "I". If this reality and self is a brain-in-a-jar simulation, I still have five fingers - the brain may not, but the brain isn't me.

Though, this is a good example of what I meant by precision of truth. "I observe five fingers" is more accurate, and there are still better phrasings, but individual inability to precisely communicate truth does not imply truth does not exist.

> Well, if you are not a brain in a jar, maybe. As far as I am concerned, I don't have any undeniable evidence that you even exist, nor that I have hands.

If I am a brain in a jar, do I have hands?

Well, then I would still have virtual/simulated/apparent hands. And, when I say "hands", by that word I mean those virtual/simulated/apparent hands. Hence, even in that scenario, I do indeed have hands.

The ordinary meaning of the English word "hands" encompasses both "ultimately physical hands" (of the kind I might have if I am not a brain in a jar and also not some AI living in a computer simulation) and also "simulated hands" (of the kind I might have if either of those hypothesises were true). So, either way, it is true that "I have hands". When I say that, I am not expressing any opinion on the truth or falsehood of those sceptical hypothesises. If I wanted to express an opinion on those sceptical hypothesises, I need to explicitly exclude them, i.e. "I have ultimately physical hands".

I believe what he meant is that all your “truths” are the same as the believe in the example. You have reasons to believe that those things are true, and certain observations made by you corroborated your hypothesis, leading you to conclude they were true.

For instance, I have 5 fingers in each of my hands: you see them, you use them, so they must exist and be real. I’m no philosopher, but I suspect their arguments would go towards how can you trust your senses and perception to prove ideas about reality...

What is your definition of truth? It seems to me like this is an issue of abstraction layers. The truth can be simply described within one layer, but adding layers requires more precise language.

It's a bit like arguing "I" don't move my arm because neural signals trigger muscles to move the arm. Both are true, the language is just more involved if we don't share a common definition of "I".

Not merely pedantic miscommunication. What do you mean by "hands"?

We understand what hands are, but imagine trying to write a program that infers what hands are, and works automatically for non-humans as well.

And then it has to work for people who don't have five fingers, too.

There are a lot of assumptions embedded in words. None of these are generally interesting, which is why they're best ignored. But they exist.

> I've slowly been converging upon the belief that there's no such thing as "truth". There are only models and predictions, and some models lead to better predictions than other models. (What then is a mathematical truth?)

This seems like saying, "The truth is that there is no truth." Or the truth is that there is a set of models which are more or less the case and this meta-model is the actual truth. In both cases, the argument seems to implicitly rest upon the truth, or a belief of the nature of truth without getting out from under the thing that the argument attempts to deny. How can we talk about there being no such thing as truth without it turning into a sentence like:

This statement is false?

Do you not believe that the international network of atomic clocks that measures the passing of time actually exists? This is all that we mean when we attach numbers to particular points in time, i.e., the "truth" that the current time is "noon" can be judged by reference to that atomic clock network.

I think the problem with this is that with no notion of truth, there is no way to test your predictions.

Sounds a bit like the pragmatic theory of Truth as proposed by William James.

https://en.m.wikipedia.org/wiki/Pragmatic_theory_of_truth

What determines whether your predictions are correct?

I have a slightly different take than you. Suppose you believe A to be true. Furthermore suppose no evidence could be presented that would convince you that A is not true. For you then, isn’t it the case that A is indeed true? It may not be true to someone else but for you that truth exists.

I guess I’d say is person and time dependent. Perhaps you agree and would say there is no such thing as objective truth?

I wouldn't dignify that with the word "truth", and I don't think you should, either ("you" meaning sykh, not "you" meaning the person in your thought experiment).

This person has a very strong belief, so strong that they put the label "truth" on it. But that's a different meaning than the word has had in this discussion thread - so different that different words should be used, to keep us straight. The version you mention here might be called "strongly-held belief" - held so strongly, in fact, that the person holding it can't imagine that it could be questioned, let alone false.

But for the rest of the comments, I think that people are using "truth" to mean "correspondence to reality". I think that that is quite different from "strongly-held belief", because I don't think that strongly-held belief changes reality.

I for instance believe that 1+1=2 under the first order Peano axioms of arithmetic. This belief of mine is so strong that it would be imposssible to convince me otherwise. This is a truth for me (and for every other sufficiently educated person). But there are differing levels of assuredness in our beliefs. Some not as strong as others. Therein lies the rub when one talks about “reality”. Isn’t reality ultimately person and perception dependent? Are there objective, universal truths?

For me a strongly held belief is more or less equivalent to truth for the person who holds such a belief.

There are varying levels of assuredness, but they also line up pretty nicely with the varying levels of experience that people have. If you have more experience verifying something to be true, then that indirectly serves as justification for the beliefs that everyone else has, even if they are not personally so sure.

The truth is a telescope of verifications that trace back through history. It's not a personal thing. It doesn't stop at your skull.

Incidentally, the only time I ever hear people saying truth is relative or doesn't exist is when those people are simply trying to promote their own social standing or take advantage of people.

The personal thing is what a person perceives as truth. Everyone has a set of beliefs that they consider true. The set isn’t the same for everyone. Isn’t it the case then that truth, at least some portion of it, is person dependent?

No, it certainly is not. That only means that one's knowledge is limited, not that the things you have knowledge of are inherently subjective.

The fact that two people disagree about something does not automatically imply that they are both wrong, only that at least one of them is.

I think what you are describing is dogma, not truth.

How is that distinguishable from truth from the perspective of the one who holds said belief?