Saturday, January 09, 2010

Are Non-Christians Unable to "Account for" Their Counting?

Presuppositionalists are often fond of quoting a statement commonly attributed to Cornelius Van Til, to the effect that
Unbelievers can count, but they cannot account for counting. (Paraphrased by Chris Bolt in his blog entry titled An Objection That Does Not Count; see also, among others, Greg Bahnsen, Van Til’s Apologetic: Readings & Analysis, p. 407)
Similarly, in his essay Counting, Infinity, and the Foundation of Knowledge, James D. Nickel concludes:
In other words, the unbeliever can count without being able to account for counting. Unbelievers can count but they cannot offer a philosophy that accounts for their practice of counting. Only the believer, redeemed by grace through Christ and in subjection to God’s written word, can truly account for the ability to count.
What I have observed as a habit among presuppositionalists who recite this viewpoint, is twofold:
a) They provide no argument to support it, and
b) They do not explain how they as Christians “account for counting” beyond merely asserting that the existence of their god is somehow a necessary precondition for this ability.
In the case of the first deficiency, note that the claim that non-Christians “cannot account for counting” is a categorically negative claim, and the burden of proving such claims is notoriously difficult to meet. Just how would one go about proving such a claim? How does the believer know that there is not some non-Christian out there who can “account for counting”? The reply “Van Til says so” certainly would not be very compelling, unless of course one were confessionally sympathetic to viewing non-Christians in a negative light. But this amounts to an attitude, not a proof. At any rate, typically the farthest that a presuppositionalist usually goes in making good on such self-serving denouncements of non-Christians, is to quote bible verses, a move which suggests that apologists have nothing better than to beg the question when it comes to supporting their assertions about non-Christians.

In his article (cited above), James D. Nickel does make a fainthearted stab at trying to provide some kind of rationale for supposing that counting presupposes the Christian god. For instance, he states:
At the very beginning of mathematical foundations, with simple counting numbers, we are introduced to a concept that transcends and perplexes human comprehension. We can conceive of the concept of infinity (through the counting numbers) only because we are made in the image of the infinite, eternal, and personal God of the Scripture.
Apparently Nickel seems to think that, since the number series is “infinite,” this means that we can only conceive of the number series (qua an infinite series) “because we are made in the image of the infinite, eternal, and personal God of the Scripture.” How precisely this is supposed to follow, is not explained.

But even Nickel points out a significant difference between the Christian god and the number series. Having just affirmed that the Christian god is eternal, Nickel explains this as follows:
By eternal, we mean “without beginning or end.” God is not subject to time.
So, according to Christianity, the Christian god is “without beginning or end.”

By contrast, however, the number series already represents a departure from the superlative characteristics attributed by Christianity to its god:
We see that in the set of natural numbers there is a dim reflection of the nature of this transcendent God. Although this set has a beginning (the number 1), it has no end.
While the Christian god is allegedly “without beginning or end,” the number series does have a beginning. It starts at a specific point, while the Christian god, as Christianity imagines it, has no starting point. Thus the number series “is a dim reflection of the nature” of the Christian god. So quickly its incorruptible brightness loses its sheen.

It is interesting to note, however, that Christianity claims that its god is “infinite.” Nickel explains:
By infinite, we mean “without limitation.” God is not subject to any limitations. He is without boundary limitations.
In other words, Christianity affirms the existence of an actual infinite. Believers, then, could not know when their god’s being stops and when another being (say, some thing it is said to have created) begins. It is “without boundary limitations,” so the believer has nothing to go on to distinguish the Christian god from anything else.

Of course, Objectivists are right in dismissing the claim that such a being exists from this very ascribed attribute, since Objectivism recognizes the fact that the actual is always finite. The axiom of identity tells us this: to exist is to be something specific, finite. If something exists, it is itself, nothing less, nothing more. If the Christian god existed, it would be just one among a gazillion other entities.

But believers are anxious to resist any such conception, so they inflate their god with puffed up imagery, and lofty descriptors to suit.

What’s even more perplexing is the fact that it never seems to occur to apologists like Nickel that the concept ‘infinity’ essentially refers nothing more than to one’s ability to continue extending a series indefinitely. It is true that the number series has no terminus; it is “infinite” in the sense that one can always continue to add more units to whichever specific number he has identified. As Ayn Rand poignantly notes:
An arithmetical sequence extends into infinity, without implying that infinity actually exists; such extension means only that whatever number of units does exist, it is to be included in the same sequence. (Introduction to Objectivist Epistemology, p. 18; emphasis added.)
Understood in this manner, the notion that the number series is “infinite” is rationally sensible, conceptually viable, and in no way implies the existence of a supernatural conscious being which is an “actual infinite.” Indeed, that apologists like Nickel seem to think that the infinity of the number series (which, as I have pointed out, is merely a potential, not an actual) is somehow a reflective indicator of the Christian god, only confirms that the Christian god is in fact merely imaginary. Indeed, we can imagine extending the number series beyond any finite number indefinitely, forever, without implying that anyone will ever undertake such a task.

In the case of the second deficiency which commonly accompanies the presuppositionalist recitation of the claim that non-Christians “cannot account for counting,” namely the failure to provide a specifically Christian “account for counting,” this is to be expected. Exactly how does Christianity “account for counting”? What you can count on here will not be a detailed explanation, but an appeal to superficial, even faulty associations, just as we saw in Nickel’s attempt to link the potential to extend the number series indefinitely to the “infinity” which Christians attribute to their god. And we saw how far that went.

What is lost in the presuppositionalist handling of the entire topic, however, is the inescapable fact that counting is a conceptual activity. But presuppositionalists do not offer a conceptual understanding of numbers and of counting in their “account for counting.” Instead, they want to point to an imaginary character from a storybook as the substance of their “account for counting.”

As an Objectivist, I “account for” counting objectively, specifically by first recognizing that numbers are a type of concept, and thus deferring to the objective theory of concepts. Human minds are certainly capable of counting. Also, Ayn Rand, in developing the objective theory of concepts (Introduction to Objectivist Epistemology), identified the fact that concept-formation is intimately related to measurement (pp. 5-18), noting that measurement as an epistemological process begins at the perceptual level of consciousness (pp. 14-15), and specifically provides an account of concepts of numbers and the process of counting in terms of the objective theory of concepts (pp. 63-64). So contrary to what Chris Bolt claims, my counting is not inconsistent with my worldview's fundamentals (since the conceptual activity of counting rests firmly on and is wholly consistent with the primacy of existence), I am in no way "borrowing" from the Christian worldview (which affirms the existence of invisible magic beings, miracles, knowledge through faith, the notion of "sensus divinitatus," a "Great Commission," the view that human beings are essentially "depraved," the belief that morality is sacrificial in nature, etc.) when I count things. Moreover, not only is there no need to point to some “supernatural” being or other figment of one’s imagination to “account for” counting, presuppositionalists – in claiming that non-Christians “cannot account for counting” – fail to interact with the Objectivist "account for counting" to which I have alluded here.

This is especially curious given the fact that some presuppositionalists do in fact acknowledge the conceptual nature of counting. For instance, Greg Bahnsen describes counting as follows:
Counting involves an abstract concept of law, universals, or order – which contradicts the unbeliever’s view of the universe as a random or chance realm of material particulars. (Van Til’s Apologetic: Readings & Analysis, p. 42n.18)
Notice how Bahnsen seeks to validate the claim that non-Christians “cannot account for counting” by assuming that, by virtue of being non-Christians, they necessarily assume the view that the universe is “a random or chance realm of material particulars.” Bahnsen continues, saying:
By rejecting God’s word, the unbeliever would not in principle be able to count and measure things. (Ibid.)
Here Bahnsen is clearly operating on a false dichotomy: either believe that the Christian god is real, or be committed to the view that the universe is “a random or chance realm of material particulars.” I explore this commonly assumed fallacy and expose its flaws in my blog The Concept of “Chance”: Right and Wrong Uses. But will presuppositionalists allow themselves to learn from this?

Of course, if a person has little or no understanding of how the human mind forms concepts, if he ignores the fact that numbers are a type of concept, and adheres to a worldview which blurs the distinction between the real and the imaginary (as Christianity does), then perhaps he might think that there’s something fundamentally supernatural involved in the process counting things. The process of counting is mysterious to him, so he assumes it is mysterious to everyone else, and uses this platform of ignorance as an opportunity to reinforce a confessional investment. In this sense, he would be essentially operating on an inference from ignorance: he does not understand what is happening in his mind when he counts things, so he figures that some invisible magic being is ultimately responsible for this phenomenon.

Now I have pointed out before that, as a worldview, Christianity suffers from a fatal deficiency just by lacking a theory of concepts (see for instance here and here). You can comb the Psalms, peruse the books of the prophets, or analyze the Pauline epistles, but you’ll find no theory of concepts proposed to guide its readers in understanding the process by which the human mind forms concepts and integrates them into larger cognitive structures. Christianity has no theory of concepts, so it cannot provide a conceptual “account for counting.” Thus it leaves believers completely in the dark on just how the human mind enumerates anything.

by Dawson Bethrick

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Blogger C.L. Bolt said...

Hi Dawson,

For some reason the links are down.

January 09, 2010 10:27 PM  
Blogger Bahnsen Burner said...

Thanks, Chris. I've corrected them now. It's one of the deficiencies of blogger's editing software - it automatically inserts the blogger domain into hyperlink tags when pasting from an external source (e.g., MS Word, which is where I tend to do my work). Very annoying, but something I have to live with apparently.

By the way, Nal, if you're out there, I tried it again and it still isn't behaving. It seems not meant to be. The editor features are there, the icon for page break is there, and the tag for the break is in my blog's coding. But it simply doesn't show up once published. Perhaps prayer may solve the problem?


January 09, 2010 10:45 PM  
Blogger NAL said...

This comment has been removed by the author.

January 10, 2010 5:49 AM  
Blogger NAL said...

Have you tried:

In the Edit HTML option for editing the post, insert:

<!-- more -->

Note spaces before and after "more".

January 10, 2010 5:50 AM  
Blogger Tim said...

Wouldn't the fact that many animals have their own ability to "count" undermine the foundation of the argument as well? It seems that counting, then, is something that is possible or has its origins in brain functions that are far less capable than a humans conceptual/abstract process. Without looking it up - I think Rand herself in ITOE explains the "crow epistemology." Animals do not "account for" their use of numbers and yet they still "count." A conceptual faculty is not a necessary condition for counting to exist.

Perhaps I am slightly off the nature of the argument?

January 10, 2010 7:34 AM  
Blogger Bahnsen Burner said...

Hello Tim,

Excellent question. Yes, Rand does discuss the “crow epistemology,” beginning on p. 62 of ITOE. While this point illustrates that rudimentary mathematical ability has a perceptual basis (she writes that the birds’ “power of discrimination did not extend beyond three units – and their perceptual-mathematical ability consisted of a sequence such as: one-two-three-many”), her overall point is in service of developing her principle of unit-economy, a benefit which only concepts can supply. She develops her point as follows:

“if we omit all conceptual knowledge, including the ability to count in terms of numbers, and attempt to see how many units… we can discriminate, remember and deal with by purely perceptual means (e.g., visually or auditorially, but without counting), we will discover that the range of man’s perceptual ability may be greater, but not much greater, than that of the crow: we may grasp and hold five or six units at most.” (ITOE, pp. 62-63)

Notice that, for Rand, “the ability to count in terms of numbers” requires conceptual ability, since numbers are a type of concept. But some quantitative differences can be perceived (such as the differences between one, two and three units). In my blog I noted that “measurement as an epistemological process begins at the perceptual level of consciousness.” Consider how a child forms the concept ‘ball’. He has before him a ping pong ball, a tennis ball and a basket ball. They have relevant similarities, but are different in size (as well as other attributes). The size differences between the units are directly perceivable. The child does not have to know what inches are, nor does he have to be able to count in terms of inches and fractions of inches, in order to recognize that one unit (the basket ball) is bigger than the other two. When he forms the concept ‘ball’, these “measurements” (for him, at this stage, they consist of “bigger” and “smaller”) as well as others are “omitted” – or, as Porter would say, “de-specified” (Ayn Rand’s Theory of Knowledge).

Incidentally, the points which Rand brings out of the “crow epistemology” plays directly into my argument that an omniscient mind would not have its knowledge in conceptual form. Rand writes:

“Since consciousness is a specific faculty, it has a specific nature or identity and, therefore, its range is limited: it cannot perceive everything at once; since awareness, on all its levels, requires an active process, it cannot do everything at once. Whether the units with which one deals are percepts or concepts, the range of what man can hold in the focus of his conscious awareness at any given moment, is limited. The essence, therefore, of man’s incomparable cognitive power is the ability to reduce a vast amount of information to a minimal number of units—which is the task performed by his conceptual faculty. And the principle of unit-economy is one of that faculty’s essential guiding principles.” (ITOE, p. 63)

In other words, the reason why concepts are so useful to our form of consciousness is because we are not omniscient, while an omniscient mind (if there could ever be such a thing) would have no need for the unit-economizing benefits of conceptual knowledge. Such a mind would presumably be one which does “perceive everything at once,” and could “do everything at once."

Hope that helps!


January 11, 2010 4:13 AM  
Blogger Bahnsen Burner said...

Hi Nal,

In Edit HTML mode, the coding you cite, namely

<!-- more -->

is there, right where I want it. But for some reason Blogger is not obeying this tag.

Incidentally, I opened the "view source" feature in IE, and noticed that the coding appears there as follows:

<a name='more'></a>

Any idea why it would show up this way in the source, and could this have anything to do with why the "read more" feature is not working? It does not show this in the Edit HTML mode.


January 11, 2010 4:21 AM  
Blogger NAL said...

Go to "Layout" -> "Edit HTML" and select "Expand Widget Templates"



After </div> add:

<b:if cond='data:post.hasJumpLink'>

<div class='jump-link'>

<a expr:href='data:post.url + "#more"'><data:post.jumpText/></a>




The line with "#more" should be one line.

I found the code here

i've also found other Minima Black templates that have functioning Jump Breaks.

January 11, 2010 4:57 AM  
Blogger NAL said...

<b:if cond='data:post.hasJumpLink'>
<div class='jump-link'>
<a expr:href='data:post.url + "#more"'>Read More</a>
</b:if >

January 11, 2010 1:30 PM  
Blogger NAL said...

It looks like you don't want:


you want the "Read More".

Whether the code snippet goes immediately after:


or after </div>, is unknown. Try both.

January 11, 2010 1:36 PM  
Blogger NAL said...

Expandable Post Summaries

Some new stuff. Good luck.

January 13, 2010 5:57 PM  
Blogger Robert Bumbalough said...

Hello Dawson and Friends

Its been almost a year since I last typed a comment, and I hope all are well and prospering. Next time I won't be such a stranger.

An interesting aspect of the topic of infinity in Christian philosophy of religion discussions is that Christians will defend the idea of an actual infinity when it comes to the scope of their gods power but will vigorously deny same in context of regression of causation. They can't have their cake and eat it too. There are serious problems for the apologists in claiming their god's Omnipotent powers are not bound by logic or the uniformity of nature. A few years ago I read a collection of essays edited by Michael Martin called "The Impossibility of God". J.L. Cowin's "The Paradox of Omnipotence" laid out insurmountable problems with the notion of Omnipotence. I wish I could recall Cowin's arguments right now. I have the book and will have to reread the essay.

On the matter of the infinite regression of causation, many non-objectivists who are atheists will strongly argue for the logical possibility of an actual infinity. Sadly, I do not recall their arguments at this time. However, I do think I recall that Cantorian Set Theory details that a Null Set has Cardinality of one and that CST stipulates that a proper subset increments its main Set's cardinality. If this is so, then an Russian Babushka Doll like infinite regression of Null Sets each containing a proper subset that is itself a Null set would have cardinality of Alph-0, yet it would be equivalent to Zero or nothing. In that sense, God would be infinite in scope and nothingness. Ha, LOL.

I would be willing to concede the logical impossibility of an actual infinity as Objectivism's argument against infinite beings would then carry the day. However, if the Christian sought to assert that infinity was an actual number then they would impale themselves on the other horn of the dilemma by accepting infinite regress of causation and thereby rejecting their god as necessarily being creator.

I'm glad I came come back here and read your stuff Dawson. Thanks for thinking and writing.

Best Wishes for 2010

Robert Bumbalough

January 19, 2010 8:38 PM  
Blogger NAL said...

The cardinality of the empty set is zero.

January 20, 2010 9:31 AM  
Blogger 小愛 said...


January 20, 2010 10:35 PM  

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