August 14, 2008

Proof Is Relative

Sometimes I wonder how well certain Biblical Scholars did in school at Math and Science. None of us is perfect, but academic logic isn’t always as consistent (or as honest) as it ought to be. Anyway, here’s my two cents on “Proof” from a Mathematical and Scientific point of view. As a High School Math Teacher, I’ll start with an example from Basic Geometry.

When doing a Formal Proof in Euclidean Geometry – and doing it properly – it’s okay to say, “Angle A is congruent to Angle B.” And a similar sounding statement, “The measure of Angle A is equal to the measure of Angle B.” is also okay. But you cannot say, “Angle A is equal to Angle B.” The mathematical reasoning behind this simple difference is to observe that shapes and objects may be nearly identical, but only values can be “equal”.

Now, most High School students rightly feel like this is nonsense, a pointless technicality. But it’s not; at least, not in proper theoretical terms. But it’s easy to see why Geometry Teachers at the High School level have different approaches on this point. Some teachers teach the difference and hold their class accountable for making both statements. (Technically, one must prove congruent angles and then also claim equivalent measures.) For convenience, many teachers allow one statement to assume the other. A case can be made for either practice, educationally, depending on the class population and postsecondary goals.

So, technically, a student has to write both statements in a formal proof. Technically, a proof without both statements is inadequate, leaving the point under debate to remain remain unproven… technically. But a student who writes one statement assuming the other IS still correct in their conclusion. They’ve just not not proved it yet… technically… because they’ve left a gap in their argument. The student’s logic requires an additional, unstated assumption.

Wphew. Got all that? Okay. Now I have two points.

One – this is just an illustration to illustrate a deeper truth, which is as follows. It is an underlying assumption of all propositional Logic that to some degree, all arguments rest on assumptions. The number of allowable assumptions is a subjective decision. We – frighteningly – WE have to decide what to assume! And oddly enough it seems there’s always someone who’s willing to question anything.

An extreme example is the always hysterical, “Do we really exist?” But everything in between that and “Yes we do!” is a sliding scale that depends purely on how much proof an individual examiner decides to require. The fact that Rene Descartes opinion was weighty and influential does not change the fact that the unpersuaded few remain so by some subjective (voluntary or involuntary) determination to demand stronger evidence.

Scientists will tell you, when pressed, that any solid theory relies on some small measure of good faith. In fact, the underlying Philosophy of Science actually suggests even basic causality is, strictly speaking, technically improvable. (Ask a Physicist sometime whether or not we can conclude that the cue ball caused the eight ball to move.)

Likewise, Integral Calculus – the practical applications of which work perfectly in the real world when accounting for all variables – is based partly on a mathematical absurdity called “The Limit”. Oh, strictly speaking in pure theory the Limit may not be considered “absurd”, but then again its existence as a concept sortof-almost-notreally requires dividing by zero, which cannot be done and therefore IS absurd. (Ask a Math Professor if you really want the details. And for real fun, ask a group of Math Professors “Does the Limit exist?” and see what happens!)

These examples bring me to my second point.

Any conclusion we draw about anything, to some degree, is based on a combination of proven and unproven “facts”. We know how chemicals react to one another, but we don’t know WHY. We know how gravity works but we don’t know WHY. We’re pretty sure (most of us) that we actually do exist, but Science cannot tell us WHY we exist.

At some point, on some aspect of any topic or debate, for whatever reasons or causes, with or without being influenced by arguments… at some point, every individual person has to eventually decide what to believe about things.

Strictly speaking, every fact and opinion one agrees with is based at least partly on faith. Objectivity is a nice ideal. But personal experience usually wins most opinions. It’s not for nothing that the saying goes, “seeing is believing.” That is, if you trust what you see! Therefore, despite the fervent desires and ostensible positions of many academics, experience remains subjective.

Oh, make no mistake – actual facts CAN be proven. It just depends on what kinds of “proof” you’re willing to accept. The Truth is not subjective, but our perceptions of it are.

Truth is not relative. But proof is.

Now, what about Christian Faith?

When the Lord Jesus Christ rose from the dead, he appeared to those who were ready to believe in him. He did not appear to the Sanhedrin. But even if they had seen him, would they have believed? Or would they have called it a hallucination? We could guess but we’ll never know. We might imagine no one could doubt their own eyes, but people see the same things all the time and make different judgments about it.

To some, the rocks and the trees are enough proof that God is. To others, no arguments will ever exist that can overcome their genuine doubt, despite their seemingly heartfelt desire to believe. I don’t know why this happens. It just does. For whatever deep, hidden and possibly even self-unaware reasons, individual human beings each hold a different standard of proof for believing in God. There is no consensus. And there is too often very little point in arguing. (Though I’m all for arguing with the right attitudes!)

Biblical Scholars ought to be more consistently logical. We both CAN and CANNOT prove the existence of God. We both CAN and CANNOT prove the reliability of certain particular statements in scripture. Just like the photon particles change direction depending on whether or not anyone’s watching, our arguments gain and lose merit depending on who’s listening. Mainly, in purely logical terms, it depends on what we allow ourselves to assume, which covers a vast sliding scale of unprovable “facts”.

All I’m saying is, we ought to acknowledge this more explicitly and give weight to respective assumptions as such, in our scholarship.

I believe God is real and the Scriptures are reliable. I choose to begin with those assumptions when reconstructing the History of New Testament Era Events. But I may or may not ever attempt to prove the truth of those assumptions to you. Why not? Because I don’t have to! Logic declares that I can merely state those assumptions as such, from the start. Everything else is still argumentatively valid.

All propositional logic relies on a certain number of assumptions. Geometry Proofs always start with a “Given”. And therefore, although Truth is not relative, proof is.

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