Pure Math, Applied Math, and A Priori Proofs
Some people think that scientists doing theoretical work can use mathematics to prove things about the real world a priori of any empirical investigation. This is wrong. Allow me to explain.
It is true that the results of pure mathematics do follow from whatever axioms one starts with a priori of any empirical observations. Indeed, empirical observations are quite irrelevant to pure math (except in an inspirational role). However, mathematics by itself cannot tell us anything about the physical world.
A concrete example will help illuminate the relationship between math and the physical sciences. Let’s consider the theory of General Relativity. GR is a mathematical model, and there are a couple of different ways to look at a mathematical model.
A pure mathematician might just be interested in the abstract “mathematical formalism” of GR. This pure mathematician could study the Einstein field equations, et cetera, irrespective of their physical significance. Indeed, an alien mathematician in an alternate universe with completely different physics could study the same mathematical formalism and derive the same results, even though the theory and results would have no physical significance in that alternate universe.
However, there’s more to a mathematical model than just an abstract formalism. A mathematical model additionally makes the empirical claim that the behavior of the real world is analogous, in a specified manner, to the behavior of its abstract formalism. This linkage between the formalism and the real world is sometimes called “the interpretation” (especially in cases where there is controversy about just how to link things up, such as quantum mechanics).
To return to the example of GR, you can derive from its mathematical formalism (in a completely abstract, a priori manner) the existence of gravitational waves. However, this most certainly does not prove that gravitational waves exist in the real world. It might turn out that there are no gravitational waves and the real world doesn’t behave in the way the model says it should. It might turn out that the model is wrong.
In light of the empirical fact that the predictions of GR have thus far agreed with experimental tests, physicists believe with fairly high certainty that GR does accurately model the behavior of the real world. We have not yet experimentally observed gravitational waves, but you’d be hard-pressed to find a physicist willing to bet money against their existence. No one, however, thinks that we’ve proved, a priori, that gravitational waves exist. If scientists did think their existence had been proved a priori, we wouldn’t bother going to the great trouble and expense of attempting to find empirical evidence for them.
In some cases, the correspondence between a mathematical model and the real world is so obvious and uncontroversial that no one bothers to explicitly lay out the formalism’s interpretation or the empirical evidence for its correctness. Consider a basket with 2 apples in it. Now toss in 2 more apples. Examine the basket, and you will find (surprise!) 4 apples. However, you cannot prove a priori that there will be 4 apples in the basket. It is an empirical question, albeit a trivial one, whether baskets of apples (which are physical things) behave in the same manner as the non-negative integers under addition (which is an abstract logical construct).
This distinction might seem hopelessly pedantic at first, but you can easily go astray by ignoring it. For example, many people naively expect photons to behave in the same manner as integers under addition, but they don’t. “Number of photons” is not a conserved quantity in the way that “number of apples” is; photons can be created/destroyed, one photon can be split into two, et cetera. Richard Feynman tells an interesting story about trying, and failing, to explain to his father how photons can do these things. I strongly suspect his father was implicitly assuming that all “particles” — a rather misleading term — must behave like apples and integers under addition. However, once you clearly understand the distinction between abstract math and empirical questions, it becomes clear that there is no a priori reason to believe that photons behave like apples or integers.
Without empirical evidence, math can’t tell us anything about the physical world. Scientists have never, ever proven anything about the real world a priori, and never will.
And your point is? Seriously, where do you think this is going? You stated that “Like Hume and Dawkins, I think that it’s pretty laughable to attempt an a priori proof of the existence of anything” and go on to claim that math isn’t *really* a priori… when Hume exempted it from his dismissal of a priori. Sorry - your demonstration that a priori pure mathematics can then be shown to be materially true is *not* a dismissal of a priori, it is a *validation* of it! Einstein’s work was purely a priori, and only a priori, until work was done to show it in the ‘real world’ - the a priori work convinced many to move past the work of Newton, after all.
What scientists do — mathematical modeling, and empirically testing the model — is very different from what theologians are trying to do in the ontological argument. We build an abstract logical construct to describe the world, and then go out and do tests to see if the world behaves as the model predicts. We don’t conclude a priori that the world must behave in the way the model predicts. Often, it doesn’t. The fact that our models can be refuted by empirical evidence should make it clear that we aren’t proving anything a priori. Furthermore, the fact that a model has been empirically refuted doesn’t mean that anyone made a mistake in the math — even if a model’s mathematical formalism is flawless, its assertion that the real world behaves like the math does can still be incorrect.
Theologians are trying to piece together a bunch of abstract logic and conclude that the world must work in a particular way (namely, that god must necessarily exist). The entire point of an a priori argument for god is to try to show that god exists without doing any empirical observations. And the only reason anyone attempts to do such a thing is because there is no empirical evidence that god exists.
Lastly, you’re just wrong about Einstein. I’ve never heard a physics professor claim he proved any physical results a priori, or read a relativity textbook which gave anything resembling an a priori proof of the theory’s correctness. If you really think all of us are wrong about this, perhaps you’d be so kind as to show us such a proof.
Jacob, I have a small suggestion - ask a theologian what we do before trying to tell us what we do. You seem to think we dit in an ivory tower, making utopias of the mind. Far from it - we are dealing with human nature, and we know it.
Einstein and his early adherents we certain - *CERTAIN* - that he described reality in his mathematics before any real world confirmations existed. Trust me - I have a minor in physics and was a physics major before I switched to theology. Indeed, his mathematics are so very good that - a priori - if they are wrong, there is something wrong with math.
The critical element is that Einstein need not do any testing or measurements to build a beuatifuly accurate picture of reality through a priori reasoning. all of your attempts to change to focus miss this point - a priori *MUST* be sound or mathematics is useless.
Ok, if theologians aren’t trying to use the ontological argument to prove that god must exist, then what are they trying to do?
A great deal of testing and measurement was necessary to arrive at the theory of relativity. The theory was specially constructed to account for experimental results in electromagnetism and, in particular, the results of the Michelson-Morley experiment. The reason Einstein and others had confidence in the theory was because it beautifully accounted for existing experimental data — the theory had empirical support from its conception. If any of them said they were 100% certain the theory was correct, then they were either overconfident, or (more likely) just speaking hyperbolically.
To be blunt: a priori reasoning can only prove things about made-up abstract entities, not real ones; math by itself is indeed quite useless.
But if you still think Einstein proved relativity correct a priori, please do give us the proof; since you have a minor in physics, I don’t think that’s asking too much.
I agree with all the other stuff you said, but Einstein always claimed that he was unaware of the Michelson-Morley experiment. The idea that the velocity of light is the same for all observers came from Maxwell’s equations.
Yes Einstein had had different conviction about theories. Briefly, in his idealistic thoughts, he referred that a real ‘Theory’ should be articulated and conceived ipso facto, without any evidence whatsoever, before the observations can corroborate the theories predictions.
In his own context Einstein you know used to devise the intense ‘Thought Experiments’, something so insightful, ideation of which can only be possible in an Einstein’s brain nerves. The slew of scientific developments taking place majorly before/after Relativity era had a different flavor: from Photoelectric effect tests to sprouting of Quantum Physics theories…you name it, all involved observation-research-theory evolution.
However, in only the case of General Relativity, there were NO experimental foundation put forth with the theory. The REAL evidences (1959+), [ not the Eddington one...that was essentially exaggerated! ] came must after theory postulation (1919), and are still coming…
This is where Einstein is marvelous and is rightly acknowledged as. The beauty of a Theory is determined by its life it lives…Newton’s ones lived three centuries. Einstein one has survived one…and counting.
His conviction was so stalwart that he sided against the Quantum worldview altogether and until his death in mid twentieth century he continued his quest for a unified theory. More the nobel prize in 1921 was given for his work on photo-electric effects, ONLY because his theory on relativity cannot be tested and evidenced by that time. Do we need more backing for this a priori thing?
In fact it has been referred that Einstein work on General Relativity during 1914-19 is a period of ‘the greatest intellectual human endeavor by a single brain’
Refer Ronald Clark’s Biography for more.
You are ignoring the enormous bodies of experiments on gravity and relativistic dynamics that existed when GR was conceived. Any theory which hoped to provide a unified account of gravity and relativity had to be compatible with all that evidence. For example, any attempted theory which failed to approximate Newton’s law of gravitation in the low-velocity limit would be immediately falsified by previous experiments.
It’s true that it took a long time to empirically confirm any of the more exotic phenomena predicted by GR. But that’s not the same thing as saying there was no experimental evidence by which to judge it. Scientists put their bets on GR because it was the only relativistic theory of gravity proposed that was compatible with existing evidence. Science doesn’t really pick out True theories; rather, it just gives us the tools to pick the best theory from extant rivals. And GR was effectively the only horse in the race.
Not a priori at all.