Argumentum ad mathematicum
Growing up as a young lad, I had an interest in mathematics. This interest waned a bit, starting in pre-calculus (doing summation the long way just doesn’t jibe well with time constraints). Nevertheless, apologetics has brought me back to studying the basics, as it were, of mathematics.
Graphic designed by Z. E. Kendall
Increasingly, those arguing for the truth of Christianity have been recognizing the value of mathematics and a mathematical perspective. Systemic design seems to ooze out of nature in the form of finite fractals—from fern fronds to broccoli to spiral galaxies. Engineered machines make use of mathematical physics equations and have parts with shapes that can be approximated through mathematical expressions. Natural biological machines appear little different. All of this beauty-produced math suggests designers for both types of machines.
But despite all of this pointing to nature and the natural world, the God debates rage on relatively uninterrupted. The concentric waves from the splash of arguments are apparently simply not going far enough across the pool of the marketplace of ideas. In a few select areas, however, a knowledge of set theory may help the apologetics community to argue more effectively for Christian or theist positions.
Set Theory and Cosmology
A knowledge of set theory may help in bringing clarity to the cosmological arguments for God’s existence. The battle in online forums against the workability of cosmological arguments sometimes includes clearly unfounded arguments. For example, one objection against the Kalam Cosmological Argument brought up in recent times has been against the premise that “everything which begins to exist has a cause for its existence.” It is argued against that point that the universe could produce itself uncaused out of nothing. However, as math professor John Lennox has mentioned, the “nothing” that physicists are speaking of concerning the quantum fluctuations capable of producing things that appear to pop into existence is not the same as philosophical “nothing” (1).
Moreover, the Kalam Cosmological Argument rests on a definition of “nothing” that is intrinsically philosophical in nature. Otherwise, Dr. William Lane Craig would not have argued (based on metaphysics) that something cannot come from nothing. As Craig has explained, even “empty space” occupied by quantum fluctuations is more than nothing, because “nothing is the absence of anything whatsoever” (2).
But does “philosophical nothing” even matter? Is it connected to anything concrete? Does the philosophy ever meet reality?
Enter set theory.
Set theory is considered to be foundational to mathematics. In some logical sense, both algebra (3) and calculus rest on its shoulders.
As Joan Bagaria stated, “The formal language of pure set theory allows [us] to formalize all mathematical notions and arguments. Thus, set theory has become the standard foundation for mathematics, as every mathematical object can be viewed as a set, and every theorem of mathematics can be logically deduced in the Predicate Calculus from the axioms of set theory”(4).
According to Liang Zongju, after Cauchy, “the most important advancement in the logical foundation of calculus was the construction of the real numbers using set theory.” Zongju argued that set theory helped with “the logical foundation of calculus” in that theories of calculus use the properties of real numbers derived from a “theory of real numbers”(5).
Robin W. Knight has recognized the notion that set theory is foundational for higher mathematics ontologically and epistemologically. The ontological foundation allows for the expression of complex structures and number sets, including real numbers. The epistemological foundation can be understood in light of the ability of mathematical questions to be reducible to questions concerning Set Theory (6).
This point about the set theory connection with algebra and calculus is important to make because modern physics stands upon the shoulders of algebra and calculus as it concerns the descriptions of reality. Consequently, one could argue that the definitions supplied by set theory may have relevance to modern science. This includes the definition of “nothing.”
The closest thing to “nothing” in set theory is the empty set. The tendency toward reductionism in set theory leads to there being only one empty set in reference to a given set, even if the given set is equal to an empty set. (This is comparable to someone recognizing that the value of zero can be added to any given algebraic term, including a term that equals zero. However, given the reductionism, we don’t go on with adding zeros to zeros ad infinitum.) The value of (absolute) zero is the value parallel to the empty set grouping.
Consequently, we end up with the rather informal philosophical statement that there can be nothing less than nothing. (This is comparable to saying that there is nothing less than absolute zero, as it concerns the quantity of anything. Absolute zero can also be viewed in terms of a calculus limit in a function. When graphed, one sees that a quantity of something already present may constantly approach absolute zero without ever reaching it.) If an entity can be conceived that is less than what we have called “nothing” and yet is not itself that “nothing,” then what we have called “nothing” cannot be “nothing” in a genuine sense. We can conceive of a “nothing” that is less than quantum fluctuations. Therefore, quantum fluctuations cannot be “nothing.” Actually, the absence of such fluctuations would get us closer to “nothing.”
In short, one could argue as follows:
Premise 1: Algebra and calculus depend on Set Theory for functional description.
Premise 2: This functional description includes absolute “nothing.”
Intermediate Conclusion 1 (IC1): Therefore, algebra and calculus depend on a Set Theory view of “nothing” (when dealing in absolute terms).
Premise 3: The philosophical definition of “nothing” is philosophically consistent with how Set Theory would view “nothing.”Intermediate conclusion 2: Therefore, given P3 and IC1, the philosophical definition of “nothing” is philosophically consistent with how algebra and calculus would view absolute “nothing.”
Premise 4: Modern physics depends on algebra and calculus for descriptions of physical reality.
Premise 5: Included in the set of options for descriptions of physical reality is the description of “nothing.”
Consequential Conclusion: Therefore, given all of the above, modern physics depends on the philosophical definition of “nothing” for descriptions of “nothing” in physical reality.
In other words, through the intersection between philosophy and mathematics that is Set Theory, the same version of nothing that undergirds philosophy must also undergird the science of physics; otherwise, we are being inconsistent either in our science or our philosophy or both.
Conclusion
Ultimately, I propose that set theory can help us to understand more. The paradigm deals principally with identity and relationships of membership (7). Consequently, set theory can help us understand the nature of basic configurations. What all this means for apologetics and theology is that set theory may have relevance to how we understand spiritual configurations as well. In later posts, I hope to elaborate on one particular way in which set theory may be helpful concerning one of those particular spiritual configurations.
As the nature of apologetics arguments become more refined and scholarly, the apologetics community is going to have to discuss matters on the levels of paradigms such as set theory. Thinking in such paradigms may lead to refining of our wording of classical arguments as well. As more scholars enter the religion debates from various fields—history, classics, law, mathematics, and others—the apologetics community will need to have more specialists and more awareness of how mathematics can support (or not support) apologetics arguments.
Bibliography:
1. Lennox, John. “QUANTUM PHYSICS IS NOTHING John Lennox 9-2014.” 27 Aug. 2014. https://www.youtube.com/watch?v=Ewbbz7cuHdw. Accessed 10 Aug. 2015. The title of this YouTube video is somewhat misleading.
2. Craig, William Lane. “Can Nothing Do Something? – William Lane Craig, PhD.” Posted 22 Aug. 2012. https://www.youtube.com/watch?v=Z8jdbWudlBM. Accessed 10 Aug. 2015.
3. This is especially the case with Boolean algebra. Dickey, Norma H., ed. “Set Theory.” Funk & Wagnalls New Encyclopedia, Vol. 23, 1990, page 316.
4. Bagaria, Joan. “Set Theory.” 8 Oct. 2014. http://plato.stanford.edu/entries/set-theory/. Accessed 10 Aug. 2015.
5. Zongju, Liang. “The Development of Calculus.” Shuxue Lishi Diangu (Historical Stories in Mathematics), Chiu Chang Publishing Company. http://db.math.ust.hk/articles/calculus/e_calculus.htm. Accessed 10 Aug. 2015. Web.
Rank and File 