Answering some questions on the theological foundations of modern science

"Answering Some Questions on the Theological Foundations of Modern Science" (James Anderson).

Chris Bolt's book is based on his doctoral dissertation "The World in His Hands: A Christian Account of Scientific Law and Its Antithetical Competitors" which is currently available to download for free via SBTS.

Handicapping the Craig/Oppy debate

I already did one brief post on the Craig/Oppy debate:

https://triablogue.blogspot.com/2020/05/does-math-point-to-god.html

I subsequently added a sentence to the first part of my two-part response. 

Now I'm going to comment on the rest of the debate. A few preliminary observations:

1. For a 70-year-old, Craig is remarkably quick on his feet, especially considering the highly abstract, technical subject-matter of the debate.

2. Oppy is a superior mind wasted on atheism. Even if atheism were true–especially if atheism were true–what's the point of mounting such a sophisticated defense of atheism? What's the point of defending atheism at all? What's the point of anything? If atheism is true, then human life is worthless, so why devote so much effort and intelligence to defending a position that renders human life worthless? Maybe Oppy doesn't view it that way, but a number of candid atheists do.

Consider defending a worldview in which it's okay to take a butcher knife and carve your mother up alive. Consider developing sophisticated arguments to defend that proposition. 

3. Moving to the meat of the debate, I think there was some miscommunication regarding Craig's statement that atheists have no explanation for the phenomena he adduces in his argument. I'm sure that's shorthand for the claim, not that atheists have no naturalistic explanations to offer, but that their naturalistic alternatives are explanatory failures. 

4. Due to time-constraints, the debate didn't have a clear-cut winner or loser, because both sides had insufficient time to expound their positions and respond to objections. Sometimes Craig had the better of the exchange, sometimes Oppy had the better of the exchange, but in some cases that's because of how the exchange abruptly ended. If each side had more time to explain their position and develop their replies, they might have a better comeback. That said, I think Craig did better overall. 

5. In the first round they got bogged down on the question of what motivates mathematicians. Here I think Craig commits an unforced error in how he formulates the first premise of his argument. That's because his formulation is overly-realiant at this point on Wigner's essay. But his argument doesn't require him to take a position on what motivates mathematicians. The issues is what's been discovered as a result of their work, regardless of their motivations. Pure math with practical applications they've developed as a result of their work, regardless of their motivations. Craig's fundamental argument is the unreasonable effectiveness of math, however mathematicians were motivated to stumble upon that insight. So Craig could rehabilitate his argument by reformulating the first premise. 

6. Initially, Craig's argument seems to hinge on scientific realism. Oppy gave examples which might support scientific anti-realism. I think Oppy had the better of that exchange. Craig needs to be able to do one of two things: (i) defend scientific realism or (ii) reformulate his argument so that it works on scientific realism and antirealism alike. 

Later in the debate Craig says the argument is about how the world appears to us. The mathematical equations allow us to describe with amazing accuracy in an uncanny number of cases the physical phenomena. Yet that seems inconsistent with earlier argument Craig and Oppy were having. But perhaps we can treat this as a clarification of the argument. 

7. On a related note, Oppy appealed to many failed theories, where the math didn't prove to be uncannily effective. Craig responded by saying the realm of math infinite while the physical world finite, so it's to be expected that in many cases the math fails to match up. I think Craig had the better of that exchange.

8. Apropos (7), I'd make an additional point. The failures that Oppy cited don't disprove the unexpected effectiveness of math (unexpected if atheism is true). Rather, they simply illustrate the fallibility of physicists. 

9. Craig appealed to the causal inertness of mathematic objects. I think Craig had the better of that exchange. 

10. Because Craig's argument appeals to laws of nature, Oppy challenged his argument on that score inasmuch as the concept or status of such laws is contested in the philosophy of science. However, when outlining his alternative to Craig's position, Oppy posting the necessity of the laws of nature. So he's faulting Craig's theistic position for a commitment which his own naturalistic alternative shares in common. Indeed, he stakes out a more ambitious claim than Craig since he regards the laws of nature as necessary whereas Craig regards the laws of nature as contingent. So that objection seems to be contradictory and self-defeating on Oppy's part. 

11. Apropos (10), while Oppy's objection was inconsistent, the lingering issue remains of whether Craig's argument is committed to some version of the laws of nature. If so, that makes his argument vulnerable at that point to disputes in the philosophy of science regarding the concept and status of such laws. It would be better if Craig could reformulate his argument so that it's not dependent on that assumption. 

Offhand, I don't see that it requires that commitment. The basic argument is that pure math has surprising empirical applications. That makes sense if the universe was "constructed on God's mathematical blueprint" (as Craig put it). It doesn't make sense if atheism is true. This might also be a way for Craig to sidestep the scientific realist/antirealist debate.

12. Oppy outlined his alternative:

A theory of modality. Every possible world shares some initial history with the actual world. Diverges from it only because chances play out differently. Those are the only possibilities that there are. The laws are necessary, the boundary conditions are necessary. Doesn't matter if you're thinking about one universe or many universe model. Where contingency comes in is the outplaying of chances. Couldn't possibly have failed not to be the case. No explaining why something is necessary. 

i) It's hard to evaluate his alternative since his presentation was so sketchy. But an acute failing of his alternative (at least as stated in the debate) is the failure to explain where the math comes from. What is Oppy's ontology of mathematics? 

ii) His commitment to nomological necessity shoulders a high burden of proof. 

iii) While it's true that once we reach necessity, that terminates further explanation, that doesn't sidestep the question of whether we rightly identified what's necessary, or what makes it necessary–in contrast to what's contingent. 

iv) What does he mean by chance? Is he alluding to quantum indeterminism? If so, there are deterministic versions of quantum theory, so he must defend his particular interpretation. 

v) Then there's his concept of the possible world. However, as commonly understood, the actual world used to be a possible world. So possible worlds don't derive from the actual world. 

In addition, from a Christian perspective, both possible worlds and the actual world derive from God. God stands behind both as their common source.

Does math point to God?

Today there was a brainiac debate between Graham Oppy and William Lane Craig:

https://www.youtube.com/watch?v=8WE1y00bwCU

I may or may not comment on other parts of the debate in a future post, but of now I'd like to zero in on a dilemma posed by Oppy:

Could God have freely chosen to make a physical world in which it was not the case that mathematical theories apply to the physical world because the structure of the physical world is an instantiation of mathematical structures described by those mathematical theories? There are two options: if not, then it seems that what you're going to end up saying is that it's necessary, that if there's a physical world, mathematical theories apply–which means you just end up with what the naturalist says. That will be the explanation. On the other hand, if it's as though it's just a brute contingency that mathematical theories apply to the physical world…because it's brutally contingent that God chose to make this world rather that other worlds that he could have made instead. When you get to free choice and you think why this rather than that, there's no explanation to be given why you ended up with one rather than the other. So it looks as though either you're going to accept the necessity or you're going to end up with ultimately it's a brute contingency. 

The answer depends on how we answer either either one of two prior questions:

i) Are mathematical structures grounded in the structure/substructure of God's mind? Does the existence of mathematical structures depend on God's existence?

ii) Is there a naturalistic mechanism to explain how the physical structure of the universe is an instantiation of mathematical structures?

Why it's hard to make a good coronavirus model

"Why It’s So Freaking Hard To Make A Good COVID-19 Model"

How bad will the coronavirus be?

Interestingly it sounds like there's something of a philosophical debate between two main camps of epidemiologists over how bad the coronavirus will be (i.e. between the "growthers" and the "base-raters"):

https://www.bloomberg.com/opinion/articles/2020-03-03/how-fast-will-the-new-coronavirus-spread-two-sides-of-the-debate

The argument from beauty and the argument from numbers

Among other things, Pruss sketches two arguments for God's existence: the argument from beauty and the argument from math:

http://alexanderpruss.blogspot.com/2020/01/wilde-lectures-now-online.html

Is 2+2=4 more certain than God's existence?

Some Christian apologists say 2+2=4 is more certain than God's existence. But is that true? 

2+2=4 may be more evident than God's existence, but is it more certain? Usually, God's existence isn't directly evident because God provides the background conditions for everything else. Of course, there are situations where God can and does make himself directly evident.

Now, it seems to be the case that 2+2=4 is a paradigm-example of a necessary truth. Nothing can be more certain than that. 

However, it's easy to imagine an evolutionary scenario in which we were arbitrarily hardwired to think 2+2=4. We can't help thinking that's the case, we can't doubt it, even if that doesn't correspond to reality. That's just how we were programmed by blind evolution.

Sure, we number things, we count things, but that's because we think they can be grouped into collections of twos and fours. But again, what if that's something we project onto physical objects (or events)?

So the deeper question is whether there's something that makes it the case that 2+2=4? And is that something God? 

I don't mean in a voluntaristic sense, as if that equation is "true" by divine fiat. Rather, mathematical structures are an aspect of God's own mind.

My objective isn't to lay out the argument for that. I'm just pointing out that as a matter of principle, God's existence may be more fundamental than mathematical equations. If so, then God's existence is more certain than mathematical equations. Their certainty is derivative. It depends on God's existence. Again, that requires an argument, and there's an argument to be had for that. 

God's infinity mirror

1. As a young boy, I remember sitting in a barbershop. I was sitting in the barber chair, having my hair cut. It was one of those neat swivel pump chairs. Behind me was a mirror all along that side of the shop. In front of me was another mirror. The combination of the two mirrors generated an infinity mirror. Sitting in the chair, I could see my reflection multiplied, receding into the never-ending distance, in ever smaller images. Boxes within boxes. 

In theory, the entire system–the boy, the mirrors, and the barbershop–could be boxes within boxes of an even larger image, like a picture on a wall. The observer could be standing outside the picture, looking at the picture of the boy in the barbershop. 

2. By the same token, I can say the Trinity is "God" as well as each person of the Deity. To take a comparison, consider an infinity mirror: An infinity mirror is a pair of parallel mirrors that generate a series of smaller scale reflections that seem to recede into an infinite distance. The same image (or information) is contained in the whole series as well as each individual reflection. In this case, unlike (1), the Persons are the mirrors.  

They're not the same Person, just as mirror-images have right-handed and left-handed orientations. There's exact one-to-one correspondence, yet a distinction remains due to chirality. The Trinity is a kind of symmetry (or the exemplar of symmetry). 

The Universe: How did it get here & why are we part of it

Roger Penrose and William Lane Craig discuss:

Progressive math

@SeaPubSchools's K-12 math ethnic studies frame work👇. I am really struggling with " who gets to say if an answer is right" in math. @kittypurrzog @wesyanghttps://t.co/yRLgeTVxFw
more info here: https://t.co/89R1bnXvVg pic.twitter.com/t6oQULPBoE

— 王启年 (@AYellowFolk) September 30, 2019

A math teacher

Francis and Edith Schaeffer, founders of the unique Christian community in Switzerland, L’Abri, had a daughter who was struggling with mathematics at her school. Priscilla had complained that she could never get through her algebra without a tutor. The Schaeffers could not afford one, so they prayed about it. The next day a Czech refugee came to visit and to ask the Schaeffers' advice about his wife’s spiritual needs. The man was very grateful. What could he do? He happened to be a math teacher! (William Edgar. Does Christianity Really Work?)

"God made the integers; all else is the work of man"

A short exchange I had with a philosophical theologian:

Josh 

Maybe "The Advancement Theory". That captures this idea of the mathematical landscape advancing in God's mind, and then also the advancement of the world, along lines without end.

Hays  

I don't see how that's supposed to work. Don't mathematical truths involve interlocking systems of entailment relations? If so, all the elements must be in place for the mathematical truth, object, or structure (however we wish to put it) to obtain. Mathematical truths can't be waiting for the coin to drop. Rather, they must exist as a unit or not at all.

Josh 

Some perhaps. But there may be basic structures that provide a foundation for iterations.

Hays

Let's put it another way, do you agree that mathematical truths are paradigm-cases of necessary truths? If so, mathematical truths can't become true. Mathematical truths can't evolve. Of course, if one's a naturalist, then one may try to reduce math to human psychology, in which case it does evolve. But that's bad theology.

Josh

A very good question. I'm toying with the idea that only basic mathematical truths are necessary (in the strongest sense).

Hays 

So you agree with Leopold Kronecker that "God made the integers; all else is the work of man".

Mathematical challenges to Darwin's theory of evolution

1. Computer scientist David Gelernter is a great mind. The interview is based on Gelernter's earlier piece. That said, see Steve's post "Giving up Darwin" for valid criticisms of Gelernter's article. Likewise David Berlinski and Stephen Meyer are highly intelligent and also make good criticisms against neo-Darwinism. Indeed, it was largely Meyer's Darwin's Doubt and Berlinski's Deniable Darwin that persuaded Gelernter on Darwinism. It's good to have all three men in dialogue like this, though I wish the interview had been longer since they covered a lot of topics that merited more time.

2. The discussion turned to intelligent design around 30 minutes or so. I think many people might find the second half of the discussion more interesting than the first half since the first half is mainly about the mathematical challenges but many people understandably find mathematics dry. In particular, it's interesting to hear two secular Jewish intellectuals, i.e., Berlinski and Gelernter, doubt intelligent design and the existence of God. The interview touched on dysteleology, theodicy, shades of anti-natalism, the argument from reason, the argument from consciousness. I think Meyer offered good if brief responses. Robinson, who is Catholic, takes on a privation theory of evil. I suspect Gelernter has in the back of mind what the unabomber did to him. I might respond to what Gelernter has said in a future post.

The Unreasonable Effectiveness of Mathematics in the Natural Sciences

Back in 1960, Eugene Wigner published a famous essay by that title. Christian apologists of a certain bent (e.g. Alvin Plantinga) appeal to this phenomenon as an argument for God's existence. For mathematical physicist Roger Penrose, the mathematical structure of the physical universe is a concrete exemplification of an abstract domain that exists outside the universe. Although Penrose is agnostic, you can see the theistic potential in that admission. Here's a recent book that provides more supporting material for that line of argument.

And here's the interview with Witten.

It's ironic that this is coming from physicists who are atheistic or agnostic. In that regard it parallels the hard problem of consciousness by secular philosophers of mind whose default position is physicalism, but acknowledge that physicalism is inadequate to account for the nature of consciousness.

The argument from numbers

However, if it’s a divine idea, then we could all be talking about the same idea, God’s idea. Divine ideas would have to be relevantly different from our own. They’re at least different in that they can secure the infinity and necessity of numbers. 

It could be that God would always have had the ideas of 2 and 4, and it could be that God would always have had the idea that 2 + 2 = 4, no matter what. But perhaps it could be that God would not necessarily have the ideas. Why should he have them? If God could have had quite different ideas (or none at all), it would be a cosmic coincidence for him to choose just those ideas, no matter what.

Let’s focus on the main subject of Leibniz’s argument first: the necessary truths of mathematics. Whenever we try to ground some domain in the divine, there’s a Euthyphro-style dilemma lurking. A comparison here between morality and mathematics might be illuminating. The traditional Euthyphro dilemma: Does God command it because it’s obligatory? Or is it obligatory because he commands it? If the former, then there’s some morality independent of God’s say-so. If the latter, then God could with as much reason command murder as he could forbid it. Does God think that 2 + 2 = 4 because 2 + 2 = 4? Or does 2 + 2 = 4 because he thinks it? If the former, then the truth is independent of God’s say-so. If the latter, then God could with as much reason have decided that 2 + 2 = 5.

Does God have the idea of 2 because it exists? Or does it exist because he has the idea? If the former, then the number is independent of God’s intellect. If the latter, then God could with as much reason never have had the idea of 2, so that it never existed.

But I wonder: Why would God’s rational nature ensure that 2 + 2 = 4? Unless there’s something intrinsically rational about 2 + 2 = 4, unless there’s something about the numbers that God’s rationality is tracking and that isn’t up to anyone, God could have dreamt up something else. But, if God’s rational nature is tracking something, then that is what mathematics is about. That is where the numbers live. It might be a luminous Platonic realm. It might be next to nothing at all. There might even be something divine about it.

http://college.wfu.edu/philosophy/wp-content/uploads/2014/08/The-Argument-From-Numbers-Ty.pdf

This poses some interesting challenges to the theistic foundations of math. 

1. I don't know what he means by saying God might not have certain ideas. 

2. Suppose we can't explain what makes mathematical truths necessarily true. That would still leave intact an argument based on God to embed mathematical infinities. 

If, likewise, numbers are ultimately mental entities, and only a timeless divine mind can ground them, then that argument remains whether or not we can explain what makes mathematical truths necessarily true. 

3. Is 2+2=4 necessarily true in isolation, or in relation to an interlocking system of mathematical relations? If the latter, then it's necessarily true within that system. 

4. Maybe there's no room for 2+2≠4 because the interlocking system is exhaustive. An infinite, transfinite totality that edges out any alternatives. Divine reason takes that as far as it can go, to ultimate internal closure. 

In order for 2+2≠4, that requires a systematic readjustment to all other equations. Maybe there can be no structure in which 2+2≠4 because there's no infinitely consistent, mutually entailing alternative. 

5. Suppose there is an alternate system, like non-Euclidean geometries. The equations would be necessarily true within those alternate system, rather than contingent or arbitrary. 

"Why you can't use logic to prove God"

I'm going to comment on this post:

http://brianhuffling.com/2018/05/01/from-whence-does-logic-come-and-why-you-cant-use-it-to-prove-god/

There is much talk about logic today. It is obviously used significantly in discussions with philosophers and mathematicians. It has also been a tool of some (particularly presuppositional) apologists to argue for God. They insist that atheists cannot account for logic since it is immaterial and universal. Since logic undeniably exists, then something else immaterial and “universal” must also exist to account for it, namely God. 

There's a lot more to the argument than: since universal, immaterial logic undeniably exists, then something else immaterial and “universal” must also exist to account for it, namely God. 

This understanding of logic is taught as if it is some ephemeral abstract notion or set of principles of reason that “exists” only in the mind with no basis in physical reality. 

The position is that logic primarily or fundamentally subsists in the infinite and timeless mind of God. God's mind is the exemplar. Logic has its "basis" in God's mind. However, math and logic are exemplified in time and space. 

That is, according to this argumentation physical reality cannot account for the principles of logic. Nothing could be further from the truth. The principles of logic, such as the principles of identity, excluded middle, and non-contradiction are not just principles of rationality. They are principles of being. Let’s look to see what they are and why they must be grounded in reality and not thought.

It's important to distinguish between divine and human thought. The position is not that logic is reducible to human psychology. It's not intuitionism (e.g. Brouwer). God's thought and God's being are conterminous. 

Yes, you could say logic is grounded in "being", but not just any kind of being will do. Physical existence won't suffice. 

The law of identity states that something is identical with itself. If a thing is “A” then it is “A”. If something is a tree, then it is a tree. This seems rather mundane and uninformative; however, try imagining reality if this were not the case. The principle of excluded middle says that something is either “A” or “non-A”. It is either a tree or a non-tree. There is no middle ground (the middle ground is excluded). The law of non-contradiction says something can’t both be “A” and “not-A” at the same time in the same sense. That is, it can’t be a tree and a non-tree simultaneously.

That confuses logic with concrete exemplifications (or property instances) of abstract objects. A tree approximates the law of identity. Ideally, logical and mathematical truths map onto static, timeless relations or objects with discrete boundaries. 

But physical objects undergo continuous incremental change. Physical objects have fuzzy boundaries in space and time. They have degrees of solidity. They exchange atoms with the surrounding environment. They blend into each other. So that comparison is counterproductive. There's never an exact match between a tree and the law of identity. 

We get our understanding of these principles from the world around us. They are not just principles of thought, but of being. The law of non-contradiction is not just that a statement can’t be both true and false. The law of non-contradiction is that something in existence can’t be and not be simultaneously in the same way. In other words, a tree can’t be a tree and not a tree at the same time in the same sense. These laws are thus grounded in being and abstracted via our knowing process. We have experience of reality and then induce said principles of being and know that they apply to all thought and experience…While these laws are undeniable and are self-evident, the source of our knowledge of them is still physical reality. 

He operates with an epistemology according to which all knowledge of universals is based on a psychological process of abstraction from particulars. 

Now, I have no problem with sense knowledge or induction. Yes, we often generalize on the basis of samples. Fine. 

But that can't be the basis of knowledge all the way down. You can't derive a concept of numbers from observing physical objects, for unless you already have numerical concepts to work with, you can't group physical objects numerically. Numbering objects requires a numerical preconception. 

You can't bootstrap logical or mathematical knowledge from sensory perception. You can't group five apples by number unless you recognize that they comprise five apples, and you're not going to arrive at that classification by staring at some apples with a blank slate mind. 

It takes knowledge to learn. It takes some prior knowledge to acquire additional knowledge. An initially empty mind has no frame of reference to evaluate sensory input. The mind of the percipient must have a logical structure which enables it to organize or reorganize sensory input. An inbuilt classification-system. 

Another way we know the laws of logic is that they are undeniable. One cannot deny something like the law of non-contradiction without using it. If one attempted to do this, he would be forced into saying that his position is true and not false, and that the opposite opinion would be false and not true. We don’t argue from more foundational principles to arrive at these principles of logic. They are first principles of thought and being. The are first because they are foundational and self-evident. They can’t be denied. Further, they don’t require, nor could they require, antecedent proof. Such proof would have to use the laws of logic.

But necessary truths of logic can't derive from contingent truths of the physical world. In many respects, the physical world might have been different. Causation is a weaker principle than logical entailment.  

Physical reality is known directly and is evident to our senses. 

Actually, physical reality is known indirectly. Physical reality is mediated to the mind via sensory perception. A process of encoded and decoded information. 

Note I said “evident” not “self-evident.” Propositions are self-evident when we know their meaning. “Bachelors are unmarried men” is a self-evident proposition because as soon as we know the meaning of the terms and the proposition as a whole, we know it is true. 

But that's different from logic. That's stipulative. True by definition. 

However, things are evident to our senses. I do not need an argument that there is a tree outside of my window. I simply see it. Thus, things are evident and the laws of logic are self-evident and undeniable. (I realize I am skipping over a veritable wonderland of skepticism and rationalism which I have no desire to deal with here. I simply don’t think I need to “justify” the existence of something I just ran my car into. If someone honestly doubts the existence of external reality, I would submit that his problem is not philosophical but psychological and he needs to seek medical treatment immediately.)

That confounds the metaphysics of math and logic with the psychology of sensory perception. 

Of course, such principles can be applied to thoughts and propositions that don’t say things about reality. Logic can be applied to fictitious beings and propositions that say something like, “All monsters live in London.” However, such fictitious beings and propositions are still based in being—that is, things that exist extra-mentally. While a fictitious being doesn’t exist in reality (by definition), we get the concepts of things like monsters from reality. In other words, following the great empirical maxim, “All knowledge is grounded in reality,” we don’t have any new ideas, even of fictitious monsters, that are not tethered to or grounded in reality.

i) Fictions, hypotheticals, and counterfactuals have their source in God's power and imagination. Something is ultimately possible because God can enact that scenario. And God's infinite imagination is the repository of all concepts. God has created rational agents with some knowledge and power. 

ii) I'd add that fictions are ideas, and therefore have a discrete identity lacking in physical objections. 

This is why the presuppositional argument for the existence of God from logic fails. A common argument from them is that atheists cannot account for logic. Logic is immaterial and universal, they say, and as such, atheists can’t account for anything that is immaterial and universal. But if what I am arguing for is true, the presuppositionalist’s argument is not successful. This is because atheists can account for logic, because logic is grounded in reality and being. Yes, God is being as such, and as “being” the laws of logic are tethered to God. (God is God, God cannot be non-God, etc.) That is, in a sense they are antecedently grounded in God because they would be the case even if the physical realm did not exist.

But that means an atheist can't account for the laws of logic inasmuch as these are essentially independent of the physical world. To be sure, some atheists are Platonic realists, but that's different from Brian's paradigm. Moreover, Platonic realism is arguably ad hoc. 

Another important note is that the laws of logic are not really immaterial. Sure the abstracted propositional form of being such as “A tree can’t be a tree and not a tree simultaneously” can be immaterial. But if logic is not merely a rational enterprise and is a second order based on the first order of physical reality, then the basis for logic is not immaterial. 

God's rationality is not a second-order exercise based on God's first-order being. That's a false dichotomy. God's mind and God's being are both first-order realities, which underlie physical reality. 

Our abstractions of the principles are mental, such as numbers, but many, if not most, philosophers do not think that numbers are real. 

That's an illicit argument from authority. Moreover, it's not coincidental that mathematicians like Quine, Gödel, and Penrose subscribe to mathematical realism. 

They like logical principles are abstracted from the real world. The number 2 does not exist. But I can say there are two trees. The two-ness is simply the addition of one more tree than the first. Math then is like logic in that the numbers are abstracted from the material world and then one can perform mental operations. But these numbers do not exist (unless one holds an extreme Platonic view). And as such, the atheist can account for logic by its foundation in sensible objects—just like he can account for numbers. Thus, the presuppositional argument for logic is going to reduce to some cosmological argument that says the universe needs a grounding in something other than itself.

If Big Ben strikes three o'clock, what do I actually hear? Do I hear three tones? No. I only hear a succession of discrete tones. I hear one tone, followed by another tone, followed by another tone. My mind apprehends three tones. That's not given in the raw stimulus, but requires an act of intellectual recognition. The mind isn't just a passive recipient of auditory input, but makes a contribution by its ability to classify the auditory input using innate mathematical categories. 

Compare a human percipient to a canine percipient. Both hear the same sounds, but only the human has the additional understanding to discern the numerical significance of the tonal sequence. A dog doesn't register "three o'clock". It lacks the intelligence to group particulars. There must be something prior in the mind to interpret what was heard as three of something.   

Christianity for Doubters

Christianity for Doubters by Granville Sewell is currently available to read for free here (pdf).

BTW, I’d say the scientific chapters are better than the theological ones.

Redeeming Mathematics

https://frame-poythress.org/wp-content/uploads/2016/01/PoythressVernRedeemingMathematics.pdf

Dawkins, Fermat, and Jesus

Richard Dawkins‏Verified account @RichardDawkins  17h17 hours ago17 hours agoMore 

Missing verse. Jesus said, no three positive integers a, b, and c shall satisfy the equation a^n + b^n = c^n. Now that would be impressive.

Apparently, Dawkins is alluding to Fermat's Last Theorem. Several issues:

i) It's prudent not to endeavor to be more clever than you are, because a failed attempt will make you look less clever than you aspire to be. 

ii) I'm no expert, but it's my impression that Dawkins bungled the formulation by omitting n>2. He should leave math to mathematicians and stick to evolutionary biology.

iii) What is the implicit argument in his tweet? I suppose it's something like this: Gospel writers can fabricate reported miracles, but a scientific or mathematic theory, theorem, conjecture, or discovery that's centuries ahead of its time can't be faked. It would be unmistakably anachronistic and undeniably impressive. Assuming that's in the ballpark of what he was gesturing at:

iv) Since modern mathematical and scientific notation didn't exist in the 1C, how could that be expressed in Aramaic or Koine Greek?

v) Since the formula would be unintelligible to Christian scribes (as well as readers), it would almost inevitably be miscopied. 

vi) Unbelievers don't think the Gospels reliably record the sayings of the historical Jesus. So even if the Gospels contained something like Fermat's Last Theorem, unbelievers could chalk that up to the narrator, or his hypothetical source, rather than Jesus. He just put that in the mouth of Jesus.

vii) Assuming (ex hypothesi) that Jesus said that, it might prove that he was a mathematical genius, but human genius is no proof of deity, and it's irrelevant to his mission as the Redeemer and eschatological judge.

viii) Is Fermat's Last Theorem especially impressive–or the solution! 

ix) Premature scientific and mathematical theories and discoveries would alter the future course of history by kickstarting math and science. It's like those scifi scenarios in which the time-traveler inadvertently changes the future because he carries his modern knowledge with him when he goes back in time, where he says or does something that seeds the past with future know-how or technology. Indeed, he becomes trapped in the past because the future from which he came no longer exists, so he's now unable to return to that point in the erstwhile timeline. 

The bell curve of theism

1. Having done a post on the bell curve of atheism:

http://triablogue.blogspot.com/2017/04/the-bell-curve-of-atheism.html

I was, not surprisingly, asked about a sequel for Christianity or theism. A few preliminaries:

i) There are different kinds of intelligence. Because Christianity is a religion centered on historical events and sacred texts, it recruits for scholars who excel at linguistics and historical reconstruction. That's a different kind of intelligence than a mathematical, philosophical, or scientific aptitude, although those are not mutually exclusive. 

To take an example, Aquinas probably had a higher IQ than Calvin. However, Calvin is interdisciplinary. A great pioneering systematic theologian. A fine philosophical theologian. An outstanding Bible commentator, by the standards of the day. He's a product of Renaissance scholarship. Knew Greek and Hebrew. It's a different skill set than Aquinas.

ii) The bell curve is about IQ. That's not a criterion of truth. A person can be very smart and very wrong. And that's a crucial distinction in theology. For instance, Rahner is very brilliant, but his frame of reference is hopelessly mistaken. This post is not a list of recommendations–although some of them I strongly recommend. 

It is useful, however, to point out that people aren't Christians or theists because they're too dumb to know any better. 

iii) My list will skimp on Jewish representatives simply because I'm more conversant with the Christian landscape than the Jewish landscape. Needless to say, Jews are disproportionately represented in math and science. There's a further distinction between nominal/secular Jews and believing Jews. 

iv) I'm not qualified to handicap how some people rank in the pecking order of science. I judge them by reputation. 

v) Who makes the cut in my divisions is somewhat arbitrary. Even within my divisions, some are more gifted than others. 

vi) I'm not sure quite how to classify pagan philosophers like Plato and Aristotle. How theistic were they? 

Likewise, I'm not sure how to classify Da Vinci. Nominal Catholic? 

vii) Finally, my lists aren't meant to be exhaustive. No doubt I may omit or overlook significant figures. Sometimes that's deliberate, sometimes inadvertent. 

2. At the very tippy top of the bell curve are some theists of genius, viz. Anselm, Aquinas, Augustine, Bayes, Berkeley, Cantor, Alonzo Church, Descartes, Jonathan Edwards, Euler, Gödel, William Hamilton, Leibniz, Maimonides, Maxwell, Newman, Newton, Pascal, Plato, Riemann, Scotus. 

That's a list of historical figures. They're about as smart as humans get. 

A bit lower on the bell curve, but very significant, are Butler, Locke, Paley, and Reid. Perhaps this is where I should put Max Planck and Werner Heisenberg. 

3. Among contemporary figures, theists at or near the very top of the bell curve probably include Francis Collins, William Dembski, Donald Knuth, David Gelernter, Saul Kripke, John Lennox, Juan Maldacena, Robert Marks, Stephen Meyer, Martin Nowak, Don Page, Jonathan Sarfati, Henry Schaefer, Rupert Sheldrake, Wesley So, James Tour, and Andrew Kamal.

That list has a focus on math, and science. And that list could no doubt be expanded. 

4. Among recent or contemporary Christian thinkers, I'd say the smartest are probably: William Lane Craig, Peter Geach, Michael Almeida, Alvin Plantinga, Vern Poythress, Nicholas Rescher, Alexander Pruss, Bas van Fraassen, and Peter van Inwagen.

(3) and (4) overlap.  

5. A list of the most talented scholars includes Dale Allison, Richard Bauckham, Gleason Archer, Roger Beckwith, F. F. Bruce, David Noel Freedman, Martin Hengel, Kenneth Kitchen, Craig Keener, Meredith Kline, John Lightfoot, Bruce Metzger, D. S. Margoliouth, Alan Millard, Adolf Schlatter, Donald Wiseman, Edwin Yamauchi, E. J. Young, Theodor Zahn. 

6. It would be a mistake to overlook artistic genius, viz. Bach, Handel, Dante. 

7. Vos and Warfield were the intellectual standouts at Princeton. 

8. Robert Adams, Elizabeth Anscombe, Alston, Swinburne, Tim & Lydia McGrew, Van Til, Wolterstorff, John Warwick Montgomery are topnotch thinkers. As are James Anderson, John Frame, Paul Helm, Paul Manata, and Greg Welty. And keep your eye on Jonathan McLatchie and Neil Shenvi.