Conversations between theists and skeptics often go like this:
Theist: “God exists.”
Skeptic: “Oh yeah? Well, prove it!”
A thorny problem lies in the term proof. What does it mean to ‘prove’ something? For example, what does it mean to prove the Pythagorean Theorem or prove that a person is not in The Matrix? Answers on the latter will vary more than the former. I have found that people tend to take the term ‘proof’ to mean various things. For example, many people think that mathematical proofs are paradigmatic. After all, it would be manifestly irrational to deny the Pythagorean Theorem once it was properly explained and demonstrated. Likewise, the skeptic often asks the theist to prove that God exists in the same way. But this unreflective position is not a particularly safe quarter.
Like many of the things we take for granted, there is a great deal of nuance regarding what it means to ‘prove’ something. Some of the greatest minds in history have addressed this question, including Aristotle, Aquinas, Russell, Whitehead, Gödel, and many others. Interestingly, some of these later thinkers, of which Frege, Hilbert, Tarski, and Turing might well be included, spent a great deal of time on mathematical proofs, axioms, and theorems. Their work is breathtaking in scope, and certainly reflects a level of genius achieved by few. This fact alone should give one pause when demanding mathematical proofs for things. Does the demand for such a proof account for the kinds of issues that must be addressed when discussing ‘proof’? Does the demand for this type of proof properly understand the framework in which one might obtain? While it is possible such depth has been accounted for, I do not think the complexity baked into the skeptical demand is usually considered.
The skeptic will generally reply that they do not need a high-level dissertation on proofs, just something simple. Prove God exists like we can prove “2 + 2 = 4.” After all, if God existed, should it really be that complicated to prove? In response, I think we should first note that complexity is not a very good test for truth. Secondly, the higher-level concepts alluded to above are still relevant. Further, such a request betrays a fundamental metaphysical assumption about numbers/mathematics which should be disclosed. Does the skeptic mean that numbers are real? Or sets? Or...? If so, then I think the type of realism implied should enter the conversation.
A skeptic holding to Platonism (perhaps implicitly) would be in a different position regarding the result and underpinning of a ‘proof’ request than a nominalist. The nominalist would be faced with the difficulty of demanding a proof based upon or like things which he does not hold as existing. In essence, the nominalist says, “prove that God exists in the same way you would prove something exists that I do not accept as really existing.” If all we are doing with mathematical proofs is rearranging the furniture of our own (exclusively) mental concepts or linguistic utterances, or winnowing word meaning conventions, then the proof demand of the theist would enter a new (and futile) light. On the other hand, the Platonist assumes abstract objects exist within his request. Thus, he essentially says, “prove that God exists in the same way you would prove that we can know about and reason with abstract objects like Two, Plus, and Four.” Given this rationale, the Platonist does not have much basis to remain in the sort of proof-skepticism from which the request springs.
What has been said so far might seem to smack of the “question the assumptions within the question” notion. This is true to a certain extent. But it is not for purposes of deflection. Instead, we must have clarification about what the interlocutor really wants. And if the interlocutor is unclear, then we risk talking past each other. If the interlocutor makes a request that is unanswerable within even his own framework, the problem remains. And if the interlocutor demands an answer to fit preconceived notions that the theist will not accept, the conversation cannot progress.
It should be said there are some theists who will not use the term ‘proof’ with regard to God’s existence. They will use arguments and present evidence to conclude, usually inductively or abductively, that God exists. To the extent the skeptic demands a proof, some theists will reply that one is not forthcoming or possible, and then explain why the skeptic should not take issue with this. It might be that the skeptic is inconsistent in his demand for proof to have sufficient knowledge for rational belief, or for other reasons altogether. Further, some theists may not even conceive of deductive arguments as ‘proofs’ because they think such a term does not apply within metaphysics. And when discussing the existence of God, we are having a metaphysical discussion. Physical scientific evidence, theories, and so forth might be used in support of premises feeding into the conclusion “God exists” but God is not in principle the proper subject of any physical science. Nonetheless, He is the subject of science (scientia = knowledge) and theists have always held that we can have knowledge of and about God.
Still, the skeptic can respond that all this mumbo-jumbo is well and good for the tweed jacket folks. But he just wants to know why he should think that God exists; he wants to know why the theist can be so (potentially) dogmatic about something so profound. Some thinkers in the past have indeed tried to prove and dissect ‘God’ by geometrical means. Spinoza is probably at the head of this class. Through a labyrinth of axioms and deductions, Spinoza lands what most commentators find to be a type of pantheism (God and nature are identified). I agree with this assessment. While occasionally presenting good food for thought, I think the Lens Grinder shows the theist what to avoid.
This leads me to try and briefly sketch some general precepts by which we might distinguish mathematical and metaphysical proofs. I think that we can hold to a notion of metaphysical ‘proof’ for God. And by ‘proof’ I mean metaphysical demonstration. Such a demonstration is deductive and yields knowledge about something. Aquinas explains it succinctly as follows
I answer that it must be said that demonstration is twofold: One which is through the cause, and is called demonstration "propter quid" [lit., 'on account of which'] and this is [to argue] from what is prior simply speaking (simpliciter). The other is through the effect, and is called a demonstration "quia" [lit., 'that']; this is [to argue] from what is prior relatively only to us (quoad nos). When an effect is better known to us than its cause, from the effect we proceed to the knowledge of the cause. And from every effect the existence of its proper cause can be demonstrated, so long as its effects are better known to us (quoad nos); because since every effect depends upon its cause, if the effect exists, the cause must pre-exist. (ST 1.2.2)
Aquinas argued that we can demonstrate God’s existence from His effects (demonstration “quia”). There was a great deal of debate in medieval philosophy about 'demonstration', how the Posterior Analytics of Aristotle should be understood, and so forth. It is beyond the scope to adjudicate all of these issues here. But I think Aquinas’ position is very defensible, and it provides a clear way forward for all concerned. I would submit that even a skeptic about God should accept this rationale, if for no other reason than to maintain a coherent position on scientific theories and many features of mathematics and logic. Among other helpful attributes, Aquinas’ rendering of demonstration can help us properly cash out mathematical versus metaphysical proofs. Here are some of the basic points:
- Mathematical demonstrations would be akin to the first type alluded to by Aquinas (above). They would be along the lines of “quid” versus “quia,” though not necessarily always so perfectly bifurcated. Mathematical proofs can become exceedingly abstract and require the supposition of axioms and precepts that are not in themselves demonstrated. In a simplified sense, a mathematical demonstration can be shown front-to-back and back-to-front, which a metaphysical demonstration might not admit of, working from front-to-back only. Though, in both cases, we can obtain knowledge (more so in the quia demonstration).
- Mathematical demonstrations are based on abstractions from real things, but the mathematician must take what these things are for granted. So, the mathematician or geometer works from reality taken as it is. But metaphysics concerns what is real, and therefore seems prior to any apprehended mathematical truths. For math and logic to work at all, there must be some objective basis. The world and intellect must be able to connect and relate. And it is just this basis that metaphysical demonstration provides to undergird mathematical demonstration. It should not be surprising that we can have “quid” demonstrations in mathematics and not for all metaphysical demonstrations.
- Quantity and relation are predicated of substance, something that exists through itself. They are accidental and cannot exist in themselves (contra Plato). Quantity and relation are necessary aspects of any mathematical activity. Thus, substance and its nature/being, etc. are antecedent to that which is predicated of it. The higher science is metaphysics, which is one of the reasons Aristotle called it “first philosophy.” Mathematics is downstream from metaphysics, and therefore mathematical demonstrations by their nature yield less science overall compared to metaphysical demonstrations. Mathematical demonstrations are by nature (pun!) narrower in scope and function.
- Mathematical demonstrations allow for more engagement of the imagination, whereas metaphysical demonstrations depart much earlier from possible imagination and revert to intellection only. At a certain point, the mathematician can more easily revert to picturing things in the mind to work out details compared to the metaphysician who often loses this recourse.
Again, this is just a brief sketch. And nothing I write here is meant to impugn or degrade mathematics in any way. The concepts of set-theory, infinity, and even numbers themselves can become exceedingly abstract, often trafficking in notions familiar to the metaphysician. In any event, mathematical and metaphysical demonstrations have important differences. We must note and discuss these differences as they apply to significant conversations, such as those on the existence of God.
If the term ‘proof’ is relegated to an impoverished conception, one that applies only in a mathematical setting, then the theist can only point out some of the issues with this and focus on the attending presuppositions. But if one understands ‘proof’ in a more robust way, such as in the medieval context of ‘demonstration’, then both sides can have a lively discussion about the veracity of the theistic argument and implications of its conclusion. The theist is not obligated to abandon the term ‘proof,’ because metaphysical demonstrations yield knowledge and describe reality, which is the work we should expect a ‘proof’ to accomplish.