Reductio ad absurdum proof of the existence of the ‘Unexplainable’ as the foundation of all knowledge

Question

First let me start with a quote around which I am posing my question:

The very ground upon which all our knowledge and science rest is the inexplicable. Therefore every explanation leads back to this, by means of middle terms more or less, as on the sea the sounding lead now finds the bottom in greater, now in lesser depths, yet ultimately must reach it everywhere. This inexplicable devolves to metaphysics.
Arthur Schopenhauer, Parerga and Paralipomena

Can we support Schopenhauer's claim metaphorically with a logic proof? For instance, don't his claims parallel using the proof by contradiction or reductio ad absurdum?

Let me try:

1. Let's for the sake of the argument say that we have some mathematical and physical explanation on why the whole world is like it is, why the Big Bang happened and everything of the form:

   A+B= L+2

2. Now, one can ask: "Ok, that explains it, but why does it look like that? Why A+B=L+2? Why not some other way?

3. Than after some time, we find another "more fundamental" equation L-2a=0 that explains the first one.

4. Repeat steps 2 and 3 ad infinitum.

So in conclusion, we can never know everything there is to know about the world so the idea that the "Unexplainable" is underneath it all holds true.

QED

While my proof is a mathematical metaphor for actual epistemological foundations, does the nature of this proof hold? Thus, are there problems with Schopenhauer claiming that metaphysical explanation must "ultimately reach... everywhere"?

Answer 1

A similar principle appears in outer set theory. Gödel attributes it to Ackermann: "The Absolute is unknowable," from which these set theorists got the concept of reflection principles (see also Horsten and Welch[16] for how those are supposed to work in justifying other principles).

The apophatique of the Absolute isn't based on a deduction in a normal sense, though, and one wonders that we would derive the underivable? We could, that is, say that there is a first-order inexplicable factoid at the nexus of first-order reality; but this observation would be second-order, and we would be called to judge that we had meta-explained the seemingly inexplicable thing beforehand. Granted, then we'd have to find the things that second-order understanding (typology) can't explain, so if we deduced what that was, we would have moved to a third-order understanding of the matter, and so on ad infinitum. But nothing would be left unexplained forever, here, except for the things we never knew to try to explain.

Also, if you have identified some X as the heart of all reality, then you at least know that about it: that X is the all-heart. But without knowing anything whatsoever else about X, how would you come to know that it was the all-heart? Calling this "the unknowable" seems like a permanent placeholder use of the description.

For a broad philosophical overview of this issue, see (from the SEP):

1. The Principle of Sufficient Reason
2. God and Other Ultimates

For an alternative formal proof, not of, "The ultimate truth is that reality is grounded in the inexplicable," but just, "There are some inexplicable truths," see about Fitch's paradox of knowability, esp. San[20].

Answer 2

You say:

So in conclusion, we can never know everything there is to know about the world so the idea that the "Unexplainable" is underneath it all holds true.

What you are asking after is known as the Principle of Sufficient Reason (SEP). At the heart of it is the potentially infinite productions of "Why?" after any explanation:

The tree grows to the sun. Why? Because it needs to use the sunlight to produce energy. Why? Because energy helps with the tree's metabolism. Why? Because photons carry energy from the sun to the chloroplast...

Note that the PSR is a controversial claim, if for no other reason than it deals with infinite processes in a physical world that is generally seen to be finite. This manifests in epistemology as the Agrippan Trilemma, where the infinite regress is one of the three horns of the bull. Note the SEP's PSR article says:

A third crucial problem for proponents of the PSR is how to address the Agrippan Trilemma between the apparently exhaustive three horns of: (i) acceptance of brute facts, (ii) acceptance of an infinite regress of explanation (or grounding), or (iii) acceptance of self-explanatory facts. Prima facie, each horn in the trilemma undermines the position of the proponent of the PSR.

As the prospect of building arguments with infinitely long chains of justification and metaphysical grounding is overwhelming to our limited intellects, philosophers (famously Descartes) gravitate towards Foundationalist Theories of Epistemic Principles (SEP). Thus, you will hear terms like "first principles", "brute facts", and "intuitions (SEP)" thrown around. Each serves the purpose of establishing the finite and beginning of the chain of reason.

You ask:

Does this proof hold?

By that you mean, does the proof show there are no foundations. But it depends on whether "after some time, we find another "more fundamental" equation" is possible. Certainly, if the universe experiences heat death, then we will stop finding equations because there is no actual infinity in that case. If thermodynamics does us in, your proof doesn't make any sense because it can't actually be constructed. But now, you are in the territory of beliefs called mathematical intuitionism. Which is an entirely different can of worms.

So, the real question is can you justify the PSR that is built into your argument? What makes you so sure that nomological facts will continue to be produced to further explain the grounding of earlier facts? What makes you so sure that you can do anything infinitely?

Answer 3

1. In the short passage from “Parerga and Paralipomena, Part II” Schopenhauer expresses a matter of course: Like the sounding lead eventually finds the point where the ground of the sea begins, also our eplanations find basic propositions which cannot be further deduced from a deeper truth.

   A philosophical argumentation cannot step down deeper and deeper for an infinite number of steps.

2. In axiomatized theories this ground is named the axioms of the theory. In philosophical argumentation one has to choose certain primordial principles or a primordial assumption which cannot be derived further.

   This assumption may be named a metaphysical assumption, e.g., the assumption that there exists a transcendental world.

I do not see the necessity to support Schopenhauer’s metaphor, which is easy to understand, by an explicit proof. And I do not understand the formal structure of your argumentation to verify that it is a valid proof.

Answer 4

This inexplicable devolves to metaphysics.

If it is inexplicable, then no amount of metaphysics will explain it.

Can we prove this logically?

No.

But logic has to start from there and so has to accommodate the inexplicable.

Can we prove this logically? Using the proof by contradiction or reductio ad absurdum?

Why R₀? Because R₁.

But why R₁? Because R₂.

Etc.

More formally:

1. R₀
2. R₁ → R₀
3. R₂ → R₁
4. Etc.

Reasoning:

Suppose R₁ explains R₀. Is R₁ the final explanation? Suppose that R₁ is the final explanation, but why R₁? Because R₂. Ah, but now R₁ is not the final explanation.

This is fallacious. It does not prove that there is no final explanation. It might be that at some point in our investigation we will find . . . the final explanation. We don't know. Perhaps R₁ is not the final explanation but R₁₀ maybe is.

Logic is essentially the implication. For example R₁ → R₀. Or, if R₁, then R₀.

This is all we have to sort out the data we have available inside our brain. But logic is not there to help us solve metaphysical problems. All it undeniably does is that it helps us survive in an unknown world. One of the mechanism of logic is that we explain things with other things. There is maybe no practical end to this process, but there is a theoretic one: reality. There is nothing outside reality, so there is nothing to explain reality with.

The rest is fair game, but reality is unexplainable.

However, maybe there is a better explanation than reality.