Absolute Truths

December 9th, 2010 by Sergio de Biasi

The truth is out there.

To anyone who has ever thought seriously about it for more than 30 seconds, I hope it is abundantly clear that the statement “there are no absolute truths” is completely ridiculous and indefensible, and should raise embarrassed laughter as soon as it’s pronounced. For those who have not thought about it for more than 30 seconds, I point out that this statement clearly self-destructs before the question “Ok, is this statement an absolute or relative truth?”. So either there are absolute truths or there aren’t, and if they do not exist, then this is in itself an (unthinkable) absolute truth, and therefore the only logically viable alternative is of course that they do exist, and we even get a bonus example: “There are absolute truths.”

But ok, this might actually seem a little too self-referential and perhaps the reader is not fully convinced of its relevance. We need better examples and criteria for what we call truth.

At the risk of being tautological, I would say that the truth is precisely what does not vary depending on what we believe.

Let us pause now to look more carefully at why this statement, instead of being what may at first glance look like an inconsequential truism, in fact contains the seeds of a fundamental philosophical concept. A tautology, in popular parlance (which follows the meaning used in rhetoric), is a statement in which we rewrite another statement using different words with the same meaning. So for example “black elephant” and “dark pachyderm” mean the same thing.  (Assume for the purposes of this argument that there is no ambiguity about the interpretation of these sentences.) So if I say that “there is a dark pachyderm if and only if there is a black elephant” I’m saying something that is patently true, in fact absolutely true, but which in a way remains completely empty of content, in the sense that it tells me nothing about the world, since strictly this is just a reaffirmation that “there is a black elephant if and only if there is a black elephant”.  Note that without resorting to any particularly mysterious concepts we’ve already run here into the existence of additional statements about whose truth we can be absolutely certain. This idea can be transformed into something entirely rigorous using mathematical logic, and if tautologies technically do not automatically lead to new assertions about properties of the world which we didn’t already know, they do at least take us to new formulations of these properties. But perhaps the reader is still dissatisfied – if that is the only type (still self-referential) of absolute truth that we are able to demonstrate, it will be difficult to go very far.

As so we arrive at a less obvious point that I want to raise. I say above “in popular parlance” because in mathematical logic a tautology is something subtly different. In mathematical logic, a tautology is something that is always true no matter what hypotheses you are starting from. So of course the logical equivalences as described above are all examples of tautologies. If I can show that the assertion X is just a way to rewrite the statement Y, then yes, “the statement X is true if and only if Y is true” is a (trivial) absolute truth. But this is not the only kind of tautology that is possible. If I say for example that “either there is a black elephant or there is no black elephant” that is also a tautology, and an absolute truth, but it is not asserting a logical equivalence. And similarly, if I say “if there is a black elephant then there is an elephant” what we have is a logical implication that is a little more sophisticated, and which is also a tautology and an absolute truth. In other words, we might know absolutely nothing about whether X is true or whether Y is true and yet be able to conclude that X implies Y is an absolute truth.

Now, notice that in all examples above we could not escape from some sort of self-reference. So here I point out that this is more or less inevitable, that unless we assume something as an axiom, as true in principle, all that we will be able to prove will be about the absolute truth of logical implications that are mandatory given certain assumptions – which we will have to refer to when drawing conclusions. And in fact, that’s all we can hope to prove with absolute certainty. Thus self-reference, or recursion, is at the core of truth, or more precisely at the core of anything we can hope to truly know for sure. This is an absolutely fundamental concept in mathematical logic, in theoretical computer science, and in philosophy: that the truths that are objectively accessible to us are exactly those which can be described recursively.

But back to the original concept. When I say that conjecture X is universally true, I’m saying that X does not depend on what assumptions you’re starting from. We’ve already concluded that such propositions do exist. The question is, how do we identify them, and will they all be trivial? (Trivial in the sense that they are all obvious or at least demonstrable).  And a bit surprisingly, when we try to deepen this concept formally, and apply it more broadly to any kind of proposition that we could state about the world, we conclude that there are truths that, although absolute – that is, true no matter which assumptions you make – it’s impossible to prove that these propositions are true! And this fact itself can be formally established as an absolute truth!

This and other considerations lead us to the need for a new word to describe logically correct conclusions – mandatory, truthful ones, which do not depend on anybody’s opinion. These are not in the general case exactly the same conclusions that can be obtained only (“tautologically” in the rhetorical sense) rewriting old statements to obtain new ones, a fact that came as a surprise to mathematicians and logicians when first shown. Of course, when we can actually rewrite a statement to obtain another, then the need of logical implication is clear. But again, there are cases where the logical implication inexorably exists but can not be demonstrated by saying the same thing with other words! That is, there  are truths that although absolute, are inaccessible to us (in the sense that although they are absolute truths we can not be absolutely sure that they are indeed absolute truths). We give to such claims (i.e. those that actually hold whether we can prove it or not) the name “valid“.

This is exactly the point where the force of the statement “the truth is precisely what does not vary depending on what we believe” shines with full force. Firstly, this would already be an interesting observation even if all truths were tautological – after all, it is not always immediately obvious when Y can be obtained by rewriting X with other words. But it goes much further than that. The fact is that even within models that are perfectly well defined and explicitly known, it is impossible to determine everything that should be necessarily true given what we believe.

Another way to put the above statement is : since absolute truth cannot depend on what we believe, then “truth is that which we are logically forced to believe when we do not assume anything a priori”.  One might hastily conclude that no conclusions can be drawn without making assumptions, but this objection has already been dismissed right at the beginning; this is clearly false.  We could then conclude a little less hastily that only trivial conclusions can be drawn from nothing, making the above sentence far less interesting though true, but the fact is that this is also not the case. The actual case is that there are truths which universally hold no matter what our assumptions are but at the same time provably there is no effective way to find out exactly what all of them are or even to determine for sure, in general, if a given statement is one of them.

On the one hand, this may seem a little daunting. But in the end, as humans, we never have direct access to what “really” is anyway and instead we only have access to what we believe. So maybe the fact that the truth does not depend on what we believe is precisely what gives us some hope for the possibility of knowing anything at all. More than that, the truth is what unites us all, in all our different beliefs, feelings, stories and missteps. Truth, not faith, revelation, tradition or instinct is what is really in common between all of our consciences, all of our individualities. Unfortunately, determining what is in fact true is extraordinarily complex, and in most cases, literally impossible.  So this whole deep unerlying existential identity that unites all beings in the universe remains latent and only occasionally perceived; meanwhile we fight to the death to defend our favorite prejudices.  And this is where we must take two steps back and transcend what we know or think we know and look at everything that is true but we can never prove or know. It remains true nonetheless, and to act as if only what we understand exists is an attitude that is securely, demonstrably universally guaranteed to be wrong.

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