Nature of Truth - in Mathematics and elsewhere

nishank gupta

Hi All,

This is my first post on these forums - so please let me know if I am violating any forum rules, and I shall try and comply as best as I can. ( I did go through guidelines).

I have had some thoughts on the nature of truth in mathematics and language, for a while now. I have tried my best to transform these thoughts into words. I have written a blog about the same on:

ng2600.blogspot.com — Nishank Gupta

Here is the first post on the subject: I have written a total of three and plan to write a couple more based on the feedback I get. ( I want to make sure I am not too far off track)

Truth in Mathematics

Things in mathematics are equally true. Does this pique your interest? If so, read on.

2 + 2 = 4, is true. So is,
sin(pi) = 0, and even if you consider a property of a random object, or a probabilistic measure, let's say,
Probability of getting a heads upon flipping a fair coin = 0.5
This is true as well.

The strange thing though is that they are equally true. We do not say that the truth of statement 1 in mathematics is more than the truth of statement 2 or so on.

I will not attempt a Tarski and define truth with a circular definition (which is inevitable), but would assume that you and me share the meaning of the word "truth".

For some reason I find this "equality in truth" to be strange and full of wonder at the same time. Do we assume that there exists something true in this world which is a constant and does not change and assign other statements of language this property?

In the realm of both the physical and the mental, certain amount of doubt can be attributed to any statement or observation or thought. I do not know for certain if anything within that set, can be said to be true with certainty. And even if it could be, as they say in language, there are no black and whites but shades of grey. But it is indeed amazing that we can do so much with one truth.

Only a sith deals in absolutes.

On the other hand, any statement is only true under certain assumptions. The truth value of some statements can only be established under certain assumptions - in the sense that the truth value of any statement A can vary between unknown and true or false under different assumptions.But, once a statement is established as true, the truth of a statement made under a stronger assumption is again equally true as the truth of a statement made under a weaker assumption.

On the uniformity of Truth - Truth is dead

There is no such thing as truth. The truth we have outside mathematics has no relationship with the truth in mathematics.

Truth from here on will refer to the truth in mathematical statements.

Making truth binary, has an interesting consequence. You now have the tool of negation. If something is not true then it is false. Normally one would leave the applications and implications to the people who work on the applied side of mathematics, but negation is a powerful tool in mathematics. There couldn't be much of math without it.

So this is how I see things right now, regarding the nature of truth:

Truth in math (calling this X, to avoid confusion) is completely different from the meaning we assign the word "True" linguistically.

X does not exist. We do not even care if it exists. But, X forms the fundamental nature of mathematics. It's more like a definition. And, once we have laid down X, we can now proceed to discuss other statements under this umbrella, with astonishing precision.

Do we lose anything when we lay down X, in the sense that does it make us unable to say a few things, which we could if we did not lay down X - I suspect that is true ( in a general sense), and maybe even inevitable, but laying down X is very powerful!. And without this X you cannot jump from a point in thought or reason to another. It is simply impossible to proceed without any assumptions whatsoever.

Yes, that does seem very close to something what an applied guy would say, but yes, this is exactly how I see things currently. X is a definition, it has no meaning outside it's scope.

Many problems are caused by confusing it (X) with the normal usage of the word "truth".

Let me know what you think of it.

Brian

Hello and welcome! Hope you enjoy your stay and feel at home here!

For some reason I find this "equality in truth" to be strange and full of wonder at the same time.

I love this sentence of your blog post by the way. It's very Greek, isn't it? The idea of philosophy as full of wonder. Such a great description of things.

I wish I knew more about mathematics which would help me answer the questions you raise about the nature of truth as it pertains to mathematics and then more generally.

The truths of mathematics seem so much different than the truths of,say, the physical sciences which are founded upon those mathematical truths. If you've read Husserl's Ideas I he has a lot to say about the distinction between the a priori truths of mathematics and his phenomenology versus the empirical truths of physics and psychology.

My primary intuition when it comes to math and something like formal logic is that the origination of these truths come from certain assumptions or axioms that we take as extremely basic and true without much in the way of justification - as Heidegger says in Being & Time, it's hard to justify or ground first principles, if not impossible.

Because mathematics is a deductive system based on those axioms about number and such, truth carries through from mathematical statement to mathematical statement via deductive reasoning that preserves the truth of those original axioms.And the derived truths are all held to be equally true, because I am not sure what could distinguish degrees of truth here.

As you say, it is amazing and full of wonder!

Does anything I said there make sense? lol

nishank gupta

↪Brian

Yes, it makes perfect sense.
Empirical truth indeed seems a little different.
But I think you are spot on regarding mathematical truth - Axioms have a property of truth and it is preserved through deductive reasoning. Yes, my point in the second post is that it is necessary to have the same property assigned to all mathematical statements - else the power in the language of mathematics would not exist.

And thank you for such a kind welcome. I admit I was waiting to be ridiculed as an amateur :)

I think the main distinction between the truth in physical sciences and mathematics is that the truth in physical sciences carry the weight of predictions, which add to their legitimacy, but at the same time those statements are based on more assumptions, one of them being that the pattern they seek to quantify, explain and predict will hold through future data.

Rich

Math is merely symbols that are carried by definition and convention. 1+1 had no meaning other than what we apply to it.

In school we are taught that one apple (the teacher displays one) plus one more apple (the teacher displays a second apple) it's equal to TWO apples and then writes it on the blackboard. And that is how convention is taught and that is the purpose of school.

Now, if course, upon inspection, we don't have two apples that are exactly the same but also by convention and practical purposes we are taught that they are the same. A child, refusing to deny individuality and uniqueness, could reasonably shout out that they are different apples and there is only one of each! Such a child is labeled a problem child and is put in special education until the child conforms to convention of society.

And in such a way, truths are developed in different cultures. It is all about consensus. There is no internet truth anywhere.

nishank gupta

↪Rich

Completely missed the point I am afraid.

Rich

↪nishank gupta

Whenever you lay down a truth, you always lose something, i.e. movement and evolution of that which is unique, and yes it does matter. Basically, there are no truths so why pretend? Let's just call them agreed conventions or consensus.

nishank gupta

↪Rich

lose something from what?

Rich

↪nishank gupta

You lose mobility of thought. Truth becomes the unchangeable anchor. Of course, the universe continues to evolve no matter what.

Best not to get involved (in personal thought) with trying to create an immobility (Truth). But if you do try (as most people do), remember the admonition of ancient philosophies (e.g. Buddhism), trying to create permanence leads to unhappiness. I guess that is why it matters.

nishank gupta

↪Rich

So you are worried about time?
You also seem to assume there was one ( thing such as truth) in the first place. Some of your sentences seem to be contradicting others.

Rich

↪nishank gupta

No assumptions anywhere. Just observations. Under inspection everything seems to be constantly changing and evolving, except in one particular situation, when I am asleep and not dreaming. But it's difficult to say 100% that such a state actually exists.

As I said, most people try to create Truths in their minds, even most practicing Buddhists, thus the admonition in the Buddhists writings, which Buddhists often ignore. As for myself, I just allow my mind to evolve mostly via the arts though I do love new philosophical ideas when presented to me, e.g. Bergson and Creative Evolution.

nishank gupta

You lose mobility of thought. Truth becomes the unchangeable anchor. Of course, the universe continues to evolve no matter what. — Rich

Do you think this is true?

Rich

↪nishank gupta

I think that it is an interesting observation and something to consider when attempting to create permanence. Allowing flow seems to create healthier environments. But the Buddhist advice is simply that though it is often called (for some reason) Noble Truths . Heck, if Truths, Immobility, and Permanence works for someone, great!

nishank gupta

↪Rich

If you had simply answered yes, you would have seen why I asked that question. :)
Anyways, later!

Rich

↪nishank gupta

For me, there are no Truths. I believe in things always subject to inevitable changes. It is a matter of how much leeway one allows in their own personal thought process. We can willfully try to create some permanence (Truths) but the energy spent on doing this may create problems. However, one can always try and observe for oneself.

nishank gupta

↪Rich

You are committing yourself to an interpretation too quickly I think.

Srap Tasmaner

↪nishank gupta

We want to be able to make inferences that rely on mathematical and empirical truth being the same, one and only, kind of truth:

If I bought 6 apples, and if I ate 1, and if you ate 1, and if 6 - 4 = 2, then there are 4 apples.

The truth of "I bought six apples" is established in one way and the truth of "6 - 4 = 2" is established in another, but a true proposition is a true proposition.

I can imagine a theory that would structure this inference completely differently, but I think we would want such a theory to generate this version of the inference as a consequence. It's just too natural and convenient to give up easily.

nishank gupta

↪Srap Tasmaner

Yes of course.
I am not proposing a different theory - things are fine just as is. In fact further down the blog I make the statements that truth needs to be a single truth/false structure to make math so powerful.

But then, the implications seem to be that this is not the same truth which we refer to in normal language.
( Still exploring and thinking on this)

I am not even 100% sure if the empirical truth and the one arrived at from deductive reasoning on axioms is the same truth. How do we establish that? Is the end result being the same enough to establish that? One works on predictions ( You predicted 4 apples will be left), and another is based on deductive reasoning. One goes beyond the physical universe as well, and is true irrespective of the boundaries of space/time. But this is not a question that I have explored yet. In common language it seems to me, that you could have a slightly different truth to statements.

Srap Tasmaner

↪nishank gupta

One way to approach the issues you raise (which I think are serious issues, very much worth thinking about) is to step back and look at assertion first. What does it mean to make an assertion? Of any kind, mathematical, empirical, metaphysical, whatever.

One idea is that the very act of assertion carries with it an idea of how the truth of the assertion could be verified or determined or established, whatever, and that's necessarily domain-dependent. An assertion is an assertion about a particular domain. So a mathematical assertion carries with it the idea of verifying the assertion by means of an effective procedure such as calculation or proof. An empirical assertion carries with it an idea about what would count as evidence for it, how that evidence could be acquired, in principle at least, etc.

Your question about inference is interesting because if you make an inference from a set of empirical premises, the result should be an empirical proposition, but when you assert the inference, you indicate that it is the validity of the inference that must be verified (conformity to inference rules, proper form of the premises, etc.). Because the result is empirical, it may be possible to disprove the result by empirical means, and then your inference becomes a reductio of one or more of the empirical premises.

Any of that make sense?

nishank gupta

I think I understand, but just to be sure...

Because the result is empirical, it may be possible to disprove the result by empirical means, and then your inference becomes a reductio of one or more of the empirical premises.

Any of that make sense? — Srap Tasmaner

You do refer to the principle behind falsification right?

So as I understand, you are saying that the act of inference from empirical evidence consists of two parts, one is the act of assertion, and the other is the inference from empirical evidence.

The way I understand is that statements or theorems in mathematics are a way of saying the same thing (assertion), but we want to explore the various ways in which we can say the same thing.
Or, to put it even more simply, we want to find out the set of all statements which have the property of truth, given a certain set of assumptions.
Empirical inferences as you say require an additional conformity to an additional set of rules.

But, in all of this process we have always assumed that the truth we obtain from mathematical statements is the same as the truth from empirical inferences. There is a singular notion of truth here, and I wanted to explore ( which is what the ideas in the blog are about), if this is indeed the case - I do not think that such is the case.

It is always a pleasure to read a well thought out argument! Thank you!

Wayfarer

in all of this process we have always assumed that the truth we obtain from mathematical statements is the same as the truth from empirical inferences. There is a singular notion of truth here, and I wanted to explore ( which is what the ideas in the blog are about), if this is indeed the case - I do not think that such is the case. — nishank gupta

I agree that empirical and a priori propositions are different in kind. However that is very unfashionable view in today's academy, where empiricism rules. The point about the kind of truth that mathematics has, is really a point about the nature of truth, and of mathematics. I think the implication is that the truth of mathematics implies that numbers have a different kind of reality to empirical objects. And that's controversial because reality is not thought to comprise 'kinds'; something is either real, or it isn't. Apples, chairs, and the number 7 are real; unicorns and the square root of two are not. End of story, most will say.

But not all. This is why there are always at least some Platonists about, especially amongst mathematicians. They too believe that number is real, but not in the same way as apples and chairs; their reality is of a different kind to the reality of objects. That's what I think you're thinking about.

nishank gupta

That's what I think you're thinking about. — Wayfarer

Nope, not really. Reality is not what I had in mind.

See, when you compare the reality of numbers to reality of empirical objects, you do so assuming that the truth in statements referring to both of them respectively is the same truth, enabling you to make the comparison. Otherwise it would be apples and oranges right?

A is true, B is true. How does the way we arrive at A compare to the way we arrive at B?
But, if I say A is T ( T for true), and B is T' ( some other form of truth), there is no simple way to make that comparison.

Wayfarer

Reality is not what I had in mind. — nishank gupta

Hey thanks for clearing that up.

Rich

You are committing yourself to an interpretation too quickly I think. — nishank gupta

I don't think so. I just believe that people tend to label as a truth what is merely an agreed consensus which it's constantly changing (and actually may be different depending upon where one lives), a consensus arrived at for practical purposes. For some reason, people seem to be always looking for truths whether it be in religion, science, or even mathematics (which in itself is nothing more than a string of symbols without any inherent meaning). It is an aspect of human consciousness that befuddles me. Why the desire to create immobility (Truths)? There is something there but I can't figure it out.

Srap Tasmaner

So as I understand, you are saying that the act of inference from empirical evidence consists of two parts, one is the act of assertion, and the other is the inference from empirical evidence. — nishank gupta

I spoke hastily and conflated a couple things, but it might actually help.

We really don't want to say that that there's the inference, and the assertion of the inference. You don't assert modus ponens when you use it. You do assert the conditional, but that too could be an empirical claim.

The issue I was reaching for is this: outside of mathematics or some other formalized domain, it's not just the validity of the inference that's up for argument, but the presentation of the issue and the applicability of the inference schema. Suppose you do something that in mathematics would be informal but entirely benign, like this:
Let x = 2
Then x² = 2x
Now imagine that argument ensues not over the second line, but over the legitimacy of saying "Let x = 2". That's how things go when you're doing philosophy, because the distinction between language and metalanguage is, well, less clear than in a formalized domain. So there's a sense in which there is always a meta-asertion that how the inference is structured is appropriate and meaningful.

Look at what you're doing here. Is it even possible to formalize the issue you're trying to address?(Note the thread next door.) I'm not sure. And there's another point there: informal reasoning is generally not quite deductive. How do you formalize analogies? Can you formalize salience? There's no harm in trying, but things are just different out here than they are in mathematics.

Now back to the nature of truth ...

Pierre-Normand

↪Srap Tasmaner

I can hear echoes of David Wiggins -- my second favorite contemporary philosopher -- in what you've just said. Is it an accident?

Srap Tasmaner

↪Pierre-Normand

Never read him, but he sounds like a pretty smart guy.

Pierre-Normand

Never read him, but he sounds like a pretty smart guy. — Srap Tasmaner

That's a clever deduction.

Pierre-Normand

↪Srap Tasmaner

Two of his papers, in particular, reinforce some of the points that you were making. Regarding the role of salience in 'sound' informal reasoning: A Sensible Subjectivim. And regarding "the presentation of the issue and the applicability of the inference schema": Deliberation and Practical Reason. Both papers are reprinted in his Needs, Value, Truth, OUP, 1998.

Srap Tasmaner

↪Pierre-Normand

Thanks!

nishank gupta

Look at what you're doing here. Is it even possible to formalize the issue you're trying to address?(Note the thread next door.) I'm not sure. And there's another point there: informal reasoning is generally not quite deductive. How do you formalize analogies? Can you formalize salience? There's no harm in trying, but things are just different out here than they are in mathematics. — Srap Tasmaner

Running a little busy, but will think about it a little more and then respond. You raise valid and interesting points.

But, even if formalizing it could be a pain, why is formal reasoning such a pillar in philosophy ( apart form the usual " concise, clear and expandable" argument), and what place does informal reasoning then have in the picture of Philosophy? You or anyone would be doing me a favor by pointing it out to me. Thanks!

I do not think formal reasoning can encompass all of available Philosophy - precision often comes with a cost.

Srap Tasmaner

↪nishank gupta

I'm for formalizing everything that can be. Maybe it's just a matter of temperament.

The place of informal reasoning in philosophy is -- Emperor? King? President for life? At least in certain areas, namely almost all of it. A few areas have been cleaned up a bit, but most of what gets really cleaned up is farmed out as a science. Philosophers are like mathematicians hanging out in the faculty lounge pre-1994 speculating about whether Fermat's last theorem is true, whether it can be proven, why it should be true, what approaches might lead to success, arguing about how much progress has been made, etc.

Nature of Truth - in Mathematics and elsewhere

Welcome to The Philosophy Forum!

Categories

More Discussions