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"The greater the sin, the greater the mercy, the deeper the death and the brighter the rebirth.” - C. S. Lewis
"This story...has the very taste of primary truth." - J. R. R. Tolkien

Monday, May 12, 2008

A Theology of Mathematics: God and Mathematics

6. Can Reality Stand Without God?

Having come to a point of reasonable certainty that numbers are real, let us briefly consider the nature of reality in general. The fact that abstract numbers exist is truly remarkable when we think about it, and it causes us to wonder why this is the case. In fact, the existence of mathematics is a good reminder of the existence of anything at all. The fact that anything exists is, quite simply, shocking! The phenomenon of existence should make us step back in awe. Not only is there a reality, but it is intelligible to us! It is essential that we not take these things for granted, and that we strive to maintain a sense of awe at reality simply because it exists.

In his paper on “Theism and Mathematical Realism,” professor of mathematics John Byl describes the three ontological realms of abstract platonic truth, physical reality, and mind. Each “emerges” from the other in a way – the physical world is fundamentally mathematical, mind is tied to the material brain, and numbers and other abstract objects emerge as mental concepts. Byl argues that from a naturalistic framework, it is very difficult to find an ontologically satisfying account of the interactions of these realms of reality (Byl 5-6). An ontological foundation that accounts for all of these realms simply cannot be satisfactorily found within these realms, so these components of reality, in and of themselves, fail to give us a reason for reality to be here, and more specifically, for this specific, contingent reality.

We began by asking “are numbers real?” and eventually concluded that they are. Now the question is “why are numbers real?” More generally, why is there something rather than nothing? Here is not the place to consider in depth so significant a question, but let us briefly examine a theistic response to it. This question essentially reduces to asking, “what could be so extraordinarily and immeasurably great – so ontologically supreme and original – that it exists of necessity and exists simply (sustaining its own being)?” From what we have seen so far, logic or numbers might seem to stand as this ontological foundation for reality, but despite their beauty and depth numbers are merely abstract realities. Mathematics may or may not take us close to the rock-solid ontological foundation of reality, but it leaves us unsatisfied. Is that really it? Is there no more at the bottom of it all than real but abstract things like numbers or beauty? Such a foundation would be too weak and nebulous – to build reality on mere abstractions would be like building a house on water. The ontological foundation we are searching for must be of greater magnitude than this. Indeed, this quality of greatness can only be true of conscious being. That conscious being is at the foundation of reality also provides a satisfactory explanation for the existence of consciousness in humans. Following this line of thinking, it becomes clear that by far the most plausible candidate for an eternal, necessarily existent, and self-sustaining thing is God.


7. God in the History of Mathematics

But where does this leave us in our quest to understand the nature of mathematics? If it is not all the way at the bottom of reality, then where exactly is it in the grand scheme of things? Before considering this question in greater detail, let us look briefly at the role of spirituality and religious sentiments in the history of science and mathematics.

Throughout the history of mathematics, many of the greatest philosophers have understood mathematics to be at the foundation of existence and deeply connected with the spiritual and the divine. This view was central to the philosophy of the Pythagoreans and Platonists. Professor of philosophy and psychology Richard Tarnas writes that for the Pythagoreans,

[T]he mathematical patterns discoverable in the natural world secreted, as it were, a deeper meaning that led the philosopher beyond the material level of reality. To uncover the regulative mathematical forms in nature was to reveal the divine intelligence itself, governing its creation with transcendent perfection and order…there exists a deeper, timeless order of absolutes (cited in Hersh 93, 96).
This connection between mathematics and the divine has continued for millennia as a mainstream concept in philosophy of mathematics. Augustine commented on similar lines that “Even if it cannot be clear to us whether number is in wisdom or from wisdom, or wisdom itself is in number and from number, or whether it can be shown that they are names for one thing; it is certainly manifest that both are true and true immutably” (cited in Hersh 105). In fact, “Many contemporary philosophers of science perceive theism to be the basis for classical mathematics and mathematical realism, both of which are therefore found to be objectionable” (Byl 1).

Even in recent decades, as naturalism and atheism have become popular in the academic circle, various thinkers such as Stephen Hawking and Albert Einstein (despite their lack of belief in an omnipotent personal being) have described the physical world with religious language and personified the laws of physics and/or mathematics as God. Quarks are named “truth” and “beauty,” and the laws of physics and mathematics are thought of as divine in a pantheistic way. Even atheists such as Richard Dawkins express wonder at the natural world and stress the importance of “truth.” Clearly, the trend of connecting science and mathematics to the idea of God or to some deeper, mysteriously spiritual reality has been a prominent theme in scientific and mathematical thought throughout history.


8. God and Mathematics

In light of this religious view of mathematics or mathematical laws, how is the Christian idea of a personal God to be brought together with mathematical realism? There are two main questions to consider here. First, does mathematics as it is exist of necessity, or could it have been otherwise? Second, did God create mathematics as an external expression of his internal being and framework for reality, or is it inherent to his being? Here I will assume a very important connection between these two questions: there is no necessary reality apart from God. Thus, anything that exists of necessity must be inherent to God’s nature. Furthermore, there is nothing in the being of God that is not a necessary reality. Thus, contingent realities, although they are ordained by the will of God, are separate from his divine nature. It is also important to approach the nature of mathematics with the theological mindset that there is nothing external to God which exists independent of God; all things exist only because God upholds their very being.


8.1. God and Logic

In approaching this issue, it will be helpful to consider the nature of logic. Orthodox Christianity has held that God is rational, acting and existing in accordance with the laws of logic. This is really the only option. In all our thinking and reasoning we must assume logic to be an accurate description of reality. Like mathematics, it is foundational and universal. To say that God created the laws of logic as entities separate from his nature or chose to structure reality to work logically implies that these laws are not true of necessity (that is, it is not essentially and necessarily true that two contradictory realities cannot both be true). This, however, goes against the most basic intuitions of human thought. The only alternative is that God himself is logical (truth and falsehood are inherent to his being), and that the laws of logic exist of necessity because God, in all his diverse yet united perfections, exists of necessity – and God exists of necessity because of who he is. In the words of philosopher and theologian John Frame, “Does God, then, observe the law of non-contradiction? Not in the sense that this law is somehow higher than God himself. Rather, God is himself non-contradictory and is therefore himself the criterion of logical consistency and implication. Logic is an attribute of God, as are justice, mercy, wisdom, knowledge” (cited in Byl 7).


8.2. Numbers Are Divine

Could the same be said of mathematical truth? That is, might mathematics exist of necessity and thus be intrinsically linked to God? Without bringing God into the picture, there is good reason to believe numbers exist of necessity. One can hardly fathom a reality where the basic numbers 1, 2, 3, etc., are somehow different or nonexistent. Our considerations of mathematical beauty provide further support for this conclusion. Still, when number is considered in and of itself, there remains some uncertainty as to whether it exists of necessity.

When the Triune God is considered alongside mathematics, however, we see a very interesting connection. If God is one being in one sense yet also three persons in another sense, how can mathematics be said to be a creation of God? The reality of numbers must be inherent in the being of the Triune God. Otherwise, how can God eternal and self-existent be said to be triune? It seems, then, that number is part of who God is just as, for example, goodness is who God is. Goodness is not a thing unto itself independent of God. Rather, goodness is defined in the being of God: God is good. In the same way, mathematics has no ontological status independent of God. Rather, mathematics is defined in the nature and being of God: God is “mathematical.” In summarizing Aquinas’s theology, John Polkinghorne says “no distinctions are to be made within God’s nature, no separation of God from God’s goodness or from God’s reason, as if they were independent constituents of the divine nature” (Polkinghorne 67). This idea of numbers or mathematical truth as part of the divine logos is very similar to the mainstream interpretation of mathematics given by theistic or Pythagorean-minded philosophers. “Augustine argued that mathematics implied the existence of an eternal, necessary, infinite Mind in which all necessary truths exist” (emphasis added, Byl 2). Nearly two thousand years ago, Nicomachus of Gerasa, who followed in the Pythagorean tradition of connecting mathematics and the divine, expresses a very similar idea:

[A]rithmetic…existed before all the others…in the mind of the creating God like some universal and exemplary plan, relying upon which as a design and archetypal example the creator of the universe sets in order his material creations…the pattern was fixed, like a preliminary sketch, by the domination of number preexistent in the mind of the world-creating God, number conceptual only and immaterial in every way, but at the same time the true and eternal essence (cited in Hersh, 94).
More recently, Alvin Plantinga has espoused the similar view that mathematical truth is linked to God’s nature and eternally present in his mind (Byl 8). Plantinga, however, seems to separate necessary truths from God, saying that they are not in his control (Zwier 4). This idea of necessary truth outside of God is problematic, though, because “if abstract objects also exist a se [of necessity], then God’s uniqueness in the universe appears now to be compromised” (Howell and Bradley 70). However, if all necessary truth is intrinsic to God’s nature, as we have assumed, then God’s uniqueness and sovereignty as the ontological source of all reality is preserved. Furthermore, if something is eternally existent in the mind of God, it would seem that it is part of God’s nature to have that thing eternally present in his thought. That is, there is a natural progression from eternality in the mind of God to ontological reality in the being of God and aseity as part of the divine essence.

Professor of philosophy Christopher Menzel agrees that “the abstract objects that exist at any given moment, as products of God’s mental life, exist because God is thinking them,” (Howell and Bradley 73) and he agrees that abstract objects like numbers exist of necessity. If we accept that the Trinity exists of necessity, then number must exist of necessity because the Trinity implies number. However, Menzel argues that objects that exist necessarily need not be uncreated or ontologically independent: “God necessarily thinks them; that is, God creates them necessarily” (Howell and Bradley 73). While necessary existence may not strictly imply uncreated, ontologically independent status, things that exist of necessity must in a sense remain grounded ontologically in the uncreated and necessarily existent being of God, as part of his nature, and if they are part of God’s nature, then they are in some sense uncreated. This idea of the necessary creation of abstract objects like numbers is, in my mind, a possible but less likely description of reality; it is more natural to place such foundational, necessary objects fully within the divine nature than it is to separate them from God while at the same time holding that they are necessarily in God’s mind and necessarily existent.

The reality of numbers and mathematical relationships is best understood in this light – as part of the ontologically essential and foundational (yet incomprehensible) unity of the diverse perfections of the Trinity. Furthermore, the beauty of mathematics is best understood in this light, that is, as the beauty of God himself. Our God is such a God that in him all the diverse tapestry of mathematics is brought together into a coherent whole with a deep beauty and sublime majesty that takes our breath away. Not only so, but there is also a unity among all the attributes of God. Mathematical truth is intrinsically tied to each facet of God and to the whole, and there is a unity in the being of God that transcends the diversity of his nature. The beauty of unity amidst diversity in mathematics is part of a greater oneness of the vast and endless perfections of God, and of the unity of three distinct persons in one being.

Incidentally, it would seem that if number and logic are both essential to the nature of God, and there is unity in this nature, then logic and number are closely tied. Thus, although we may not be able to epistemologically reduce math to logic, there may be a deeper connection beyond our knowledge, and the logicists may have been on the right track in seeing a tie between number and logic.


8.3 Two Objections

One might object that if mathematics and logic are necessarily true or real, God is somehow constrained or limited. In response to this, it must first be noted that without its identity being ontologically grounded in the being of God, mathematical and logical truth would lose its source for existence along with all else. God is the great ontological rock at the foundation of reality; without his supreme ontological greatness, reality would have no source of being and would fall out of existence. Thus, mathematical realism loses it ground without theism. Hersh seems to realize this and writes that “The trouble with today’s Platonism is that it gives up God, but wants to keep mathematics a thought in the mind of God” (Hersh 135). Byl points out that the decline in theism in the 20th century led mathematicians to seek a foundation for mathematics in “self-evident axioms” (Byl 2), thus moving the search for a foundation for mathematics from ontology to epistemology, which, I would argue, is a fruitless endeavor. Numbers cannot exist as independent free-floating entities anymore than the justice of God exists as an abstract reality independent of God. Thus, mathematical truth cannot be separated from God. Each facet of the divine being of God only exists of necessity because the united whole of the being of God exists of necessity. Because neither mathematics nor logic itself have any reality external to God or independent of God, God is not bound or constrained in any way by logic or mathematical truth.

One might also object that sliding mathematics into the divine nature would put us on a slippery slope towards pantheism. However, other entities of beauty, such as the laws of physics, need not follow. It is only because of the absolutely universal and fundamental truth of mathematics that one would expect it to be true of necessity. Again, a Triune God cannot be conceived of without the number three. Indeed one could hardly conceive of it not existing of necessity (that is, numbers being created entities). Furthermore, we are comfortable believing that similarly abstract entities of beauty, such as absolute moral values, love, and goodness, are foundationally divine and find their grounds for existence in God. Numbers are comparable to these truths in beauty and necessity of existence and, therefore, are probably divine as well.


9. Epistemology and Revelation

If this is the nature of the being of God, what implications might there be for theological revelation and our knowledge of God? Barker asks of abstract objects, “If they are intangible, immaterial entities, how then can we have knowledge of them, and how can they be so important in our thinking?” (Barker 69). Perhaps we will never fully understand how abstract objects independent of mind and matter are made known to minds, which are linked to matter and yet are something more. However, we can, from a Christian perspective, say something of why God gave us this knowledge.

As Jonathan Edwards wrote in his great work The End for Which God Created the World, creation, in general, should be viewed in part as an overflowing of the fullness of the perfections of God, somewhat like a fountain (cited in Piper 165). Thus, the knowledge of mathematical truth and beauty in the minds of those made in the image of God is not only revelation of the divine nature to us, but also an overflowing of the nature of God into lesser creatures made in his image. Similarly, the mathematical structure of the physical world is the result of the action of a Creator who imbues all things with his likeness, in some way or another. Furthermore, in making the laws of physics and the patterns of this world so thoroughly mathematical, God is making himself known to his creatures and displaying his glory. He is saying to us, “look! – this is a small part of what I, your Maker, am like.” This revelatory understanding of epistemology and of creation as an expression of God’s character is confirmed in Scripture. In the words of the apostle Paul, “For since the creation of the world God’s invisible qualities, his eternal power and divine nature, have been clearly seen, being understood from what has been made” (Romans 1:20). Jonathan Edwards writes of the glory of God on a similar note:

The thing signified by the that name, the glory of God, when spoken of as the supreme and ultimate end of all God’s works, is the emanation and true external expression of God’s internal glory and fullness…The emanation or communication of the divine fullness…has relation to God as its fountain, as the thing communicated is something of its internal fullness…The refulgence shines upon and into the creature, and is reflected back to the luminary. The beams of glory come from God, are something of God, and are refunded back again to their original. So that the whole is of God, and in God, and to God; and he is the beginning, and the middle, and the end (cited in Piper 242, 247).


10. Conclusion

To sum up, we have examined the various philosophies of mathematics, concluding that realism offers, in general, the best account of observation and intuition. The perception of mathematical beauty provides strong support for this interpretation. Remarkably, we perceive real things beyond the physical world. Mathematical realism, however, is difficult to uphold without theism; more generally, the idea of God is by far the most plausible candidate for an ontological foundation to reality and an explanation of why anything exists. Indeed, mathematical realists throughout history have leaned towards theism and have thought of mathematical truth as almost divine. The foundational nature of mathematics, perception of mathematical beauty, and the triune nature of God suggest that mathematical truth exists of necessity in the being of God, as part of the diverse yet united perfections God. Thus, theism in turn supports realism by providing an ontological foundation. Finally, if numbers are divine rather than created, then our interest in mathematics and in the physical world as revelation of God should increase all the more. The theme of beauty and perfection in unity amidst diversity (and perhaps also of simplicity beneath complexity) is a key idea, both with respect to mathematics and with respect to God. As Ravi Zacharias has said, God is able to take the sublime and make it simple, and he is able to take the simple and make it sublime. I close with two excerpts from Scripture expressing the beauty and majesty of the revealed perfections of the divine nature.

And these are but the outer fringe of his works; how faint the whisper we hear of him! Who then can understand the thunder of his power? – Job 26:14

Oh, the depths of the riches of the wisdom and knowledge of God! How unsearchable his judgments, and his paths beyond tracing out!...For from him and through him and to him are all things. To him be the glory forever! Amen. – Romans 11:33, 36



Bibliography

Barker, Stephen. Philosophy of Mathematics. Prentice-Hall: Englewood Cliffs NJ, 1964.

Bell, Eric Temple. Men of Mathematics. Simon and Schuster: New York, 1986.

Byl, John. “Theism and Mathematical Realism.” Journal of the Association of Christians in the Mathematical Sciences (2001). <http://acmsonline.org/Byl-realism.pdf>.

Carter, Benjamin M. “The Limitations of Mathematics in Assessing Causality.” Perspectives on Science and Christian Faith 57.4 (2005): 279-283.

“Euler's Identity.” Wikipedia. 23 Apr. 2008. <http://en.wikipedia.org/wiki/Euler>.

Hersh, Reuben. What Is Mathematics, Really? Oxford University Press: New York, 1997.

Holmes, Arthur F. “Wanted: Christians Perspectives in the Philosophy of Mathematics.” Journal of the Association of Christians in the Mathematical Sciences (1997). <http://acmsonline.org/Holmes77.pdf>.

Howell, Russell W. “Does Mathematical Beauty Pose Problems for Naturalism?” Christian Scholar’s Review 35.4 (2006): 493-504.

Howell, Russell W., and W J. Bradley, eds. Mathematics in a Postmodern Age: A Christian Perspective. Grand Rapids: William B. Eerdmans Company, 2001.

Kasner, E., and Newman, J. Mathematics and the Imagination. Bell and Sons, 1949.

Piper, John. God’s Passion for His Glory. Crossway Books: Wheaton IL, 1998.

Polkinghorne, John. Science and Theology: An Introduction. Minneapolis: Fortress Press, 1998.

Nahin, Paul J., Dr. Euler's Fabulous Formula: Cures Many Mathematical Ills. Princeton University Press: Princeton, 2006.

Russell, Bertrand. Mysticism and Logic. Dover Publications, 2004.

White, Charles E. “God by the Numbers: Coincidence and Random Mutation are Not the Most Likely Explanations for Some Things.” Christianity Today Mar. 2006. 13 Apr. 2008. <http://www.christianitytoday.com/ct/2006/march/26.44.html>.

Wigner, Eugene. “The Unreasonable Effectiveness of Mathematics in the Natural Sciences.” Communications in Pure and Applied Mathematics 13.1 (1960). <http://www.dartmouth.edu/~matc/MathDrama/reading/Wigner.html>.

Zwier, Paul. “A Comparative Study of Christian Mathematical Realism and its Humanistic Alternatives.” Journal of the Association of Christians in the Mathematical Sciences (1997). <http://acmsonline.org/Zwier83.pdf>.

2 comments:

  1. Have you ever read "Contact" from Carl Sagan? Reading your blogs makes me think of his writing a bit. Of course you come to a different conclusion than he does about God. Anyway, he used transcendental numbers as evidence of an intelligent creator. I thought it was an interesting idea coming from a non-believer.

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  2. Interesting, I'll have to check it out. Recently there were some blog posts on biologos.org about a movie based on that book.

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