The 'Spirituality' of Mathematics
An atheist recently claimed, in response to my observation that mathematics must ultimately be based in God, that:
"maths just is"
Well, contrary to this commonly held belief that 'maths just is', the belief that 'maths just is' is now demonstrably false. First to be noted, there is a profound epistemological mystery as to why our minds should even be able to grasp and understand reality through the enterprise of mathematics in the first place:
Epistemology – Why Should The Human Mind Even Be Able To Comprehend Reality? – Stephen Meyer - video – (Notes in description)
"You find it strange that I consider the comprehensibility of the world (to the extent that we are authorized to speak of such a comprehensibility) as a miracle or as an eternal mystery. Well, a priori, one should expect a chaotic world, which cannot be grasped by the mind in any way.. the kind of order created by Newton's theory of gravitation, for example, is wholly different. Even if a man proposes the axioms of the theory, the success of such a project presupposes a high degree of ordering of the objective world, and this could not be expected a priori. That is the 'miracle' which is constantly reinforced as our knowledge expands."
Albert Einstein - Goldman - Letters to Solovine p 131.
The Unreasonable Effectiveness of Mathematics in the Natural Sciences - Eugene Wigner - 1960
Excerpt: ,,certainly it is hard to believe that our reasoning power was brought, by Darwin's process of natural selection, to the perfection which it seems to possess.,,,
It is difficult to avoid the impression that a miracle confronts us here, quite comparable in its striking nature to the miracle that the human mind can string a thousand arguments together without getting itself into contradictions, or to the two miracles of the existence of laws of nature and of the human mind's capacity to divine them.,,,
The miracle of the appropriateness of the language of mathematics for the formulation of the laws of physics is a wonderful gift which we neither understand nor deserve. We should be grateful for it and hope that it will remain valid in future research and that it will extend, for better or for worse, to our pleasure, even though perhaps also to our bafflement, to wide branches of learning.
Second, in the last century, Godel showed mathematics to be 'incomplete':
Kurt Gödel – Incompleteness Theorem – video
THE GOD OF THE MATHEMATICIANS - DAVID P. GOLDMAN - August 2010
Excerpt: we cannot construct an ontology that makes God dispensable. Secularists can dismiss this as a mere exercise within predefined rules of the game of mathematical logic, but that is sour grapes, for it was the secular side that hoped to substitute logic for God in the first place. Gödel's critique of the continuum hypothesis has the same implication as his incompleteness theorems: Mathematics never will create the sort of closed system that sorts reality into neat boxes.
Taking God Out of the Equation – Biblical Worldview – by Ron Tagliapietra – January 1, 2012
Excerpt: Kurt Gödel (1906–1978) proved that no logical systems (if they include the counting numbers) can have all three of the following properties.
1. Validity . . . all conclusions are reached by valid reasoning.
2. Consistency . . . no conclusions contradict any other conclusions.
3. Completeness . . . all statements made in the system are either true or false.
The details filled a book, but the basic concept was simple and elegant. He summed it up this way: “Anything you can draw a circle around cannot explain itself without referring to something outside the circle—something you have to assume but cannot prove.” For this reason, his proof is also called the Incompleteness Theorem.
Kurt Gödel had dropped a bomb on the foundations of mathematics. Math could not play the role of God as infinite and autonomous. It was shocking, though, that logic could prove that mathematics could not be its own ultimate foundation.
Christians should not have been surprised. The first two conditions are true about math: it is valid and consistent. But only God fulfills the third condition. Only He is complete and therefore self-dependent (autonomous). God alone is “all in all” (1 Corinthians 15:28), “the beginning and the end” (Revelation 22:13). God is the ultimate authority (Hebrews 6:13), and in Christ are hidden all the treasures of wisdom and knowledge (Colossians 2:3).
In other words, the truthfulness of any given mathematical equation is not found within the equation itself, but the truthfulness of any given mathematical equation, and indeed of all of math, must be derived from a source outside of the equation(s). Moreover, being that mathematical equations are completely transcendent of any space-time constraints, (i.e. mathematical equations are always true no matter what part of the universe you are in and are true regardless of whatever year it happens to be), then this outside source (cause) that guarantees the truthfulness of any mathematical equation must also be transcendent of any space-time constraints.
Also of note, Godel's incompleteness theorem is hardly the only line of argumentation in this line of thought:
Not Understanding Nothing – A review of A Universe from Nothing – Edward Feser - June 2012
Excerpt: A critic might reasonably question the arguments for a divine first cause of the cosmos. But to ask “What caused God?” misses the whole reason classical philosophers thought his existence necessary in the first place. So when physicist Lawrence Krauss begins his new book by suggesting that to ask “Who created the creator?” suffices to dispatch traditional philosophical theology, we know it isn’t going to end well. ,,,
,,, But Krauss simply can’t see the “difference between arguing in favor of an eternally existing creator versus an eternally existing universe without one.” The difference, as the reader of Aristotle or Aquinas knows, is that the universe changes while the unmoved mover does not, or, as the Neoplatonist can tell you, that the universe is made up of parts while its source is absolutely one; or, as Leibniz could tell you, that the universe is contingent and God absolutely necessary. There is thus a principled reason for regarding God rather than the universe as the terminus of explanation.
But of more interest as to drawing out the 'spirituality of mathematics', and refuting the 'maths just is' conception of mathematics, it is worthwhile to focus in on the Schroedinger equation:
Finely Tuned Big Bang, Elvis In The Multiverse, and the Schroedinger Equation – Granville Sewell – audio
At the 4:00 minute mark of the preceding audio, Dr. Sewell comments on the ‘transcendent’ and ‘constant’ Schroedinger’s Equation;
‘In chapter 2, I talk at some length on the Schroedinger Equation which is called the fundamental equation of chemistry. It’s the equation that governs the behavior of the basic atomic particles subject to the basic forces of physics. This equation is a partial differential equation with a complex valued solution. By complex valued I don’t mean complicated, I mean involving solutions that are complex numbers, a+bi, which is extraordinary that the governing equation, basic equation, of physics, of chemistry, is a partial differential equation with complex valued solutions. There is absolutely no reason why the basic particles should obey such a equation that I can think of except that it results in elements and chemical compounds with extremely rich and useful chemical properties. In fact I don’t think anyone familiar with quantum mechanics would believe that we’re ever going to find a reason why it should obey such an equation, they just do! So we have this basic, really elegant mathematical equation, partial differential equation, which is my field of expertise, that governs the most basic particles of nature and there is absolutely no reason why, anyone knows of, why it does, it just does. British physicist Sir James Jeans said “From the intrinsic evidence of His creation, the great architect of the universe begins to appear as a pure mathematician”, so God is a mathematician to’.
i.e. the Materialist is at a complete loss to explain why this should be so, whereas the Christian Theist presupposes such ‘transcendent’ control of our temporal, material, reality,,,
In the beginning was the Word, and the Word was with God, and the Word was God.
Of note: 'The Word' in Greek is Logos. Logos is the root word from which we derive our modern word 'logic'.
But the mystery of the Schroedinger equation goes even deeper to reveal 'the spirituality of mathematics'.
Wheeler's Classic Delayed Choice Experiment:
Excerpt: Now, for many billions of years the photon is in transit in region 3. Yet we can choose (many billions of years later) which experimental set up to employ – the single wide-focus, or the two narrowly focused instruments. We have chosen whether to know which side of the galaxy the photon passed by (by choosing whether to use the two-telescope set up or not, which are the instruments that would give us the information about which side of the galaxy the photon passed). We have delayed this choice until a time long after the particles "have passed by one side of the galaxy, or the other side of the galaxy, or both sides of the galaxy," so to speak. Yet, it seems paradoxically that our later choice of whether to obtain this information determines which side of the galaxy the light passed, so to speak, billions of years ago. So it seems that time has nothing to do with effects of quantum mechanics. And, indeed, the original thought experiment was not based on any analysis of how particles evolve and behave over time – it was based on the mathematics. This is what the mathematics predicted for a result, and this is exactly the result obtained in the laboratory.
"Thus one decides the photon shall have come by one route or by both routes after it has already done its travel"
John A. Wheeler
Alain Aspect speaks on John Wheeler's Delayed Choice Experiment - video
Genesis, Quantum Physics and Reality
Excerpt: Simply put, an experiment on Earth can be made in such a way that it determines if one photon comes along either on the right or the left side or if it comes (as a wave) along both sides of the gravitational lens (of the galaxy) at the same time. However, how could the photons have known billions of years ago that someday there would be an earth with inhabitants on it, making just this experiment? ,,, This is big trouble for the multi-universe theory and for the "hidden-variables" approach.
Wheeler's Delayed Choice Experiment - 2010
Excerpt: The Delayed Choice experiment changes the boundary conditions of the Schrodinger equation after the particle enters the first beamsplitter.
But why should a mathematical equation even care when I decide to implement boundary conditions to look at a particle? Mathematical equations can't care about anything! Only God can care if and when I decide to look at any particular particle!
In fact, 'the spirituality of mathematics' has now been revealed to a even deeper level through recent quantum entanglement experiments. The foundation of quantum mechanics within science is now so solid that researchers were able to bring forth this following proof from quantum entanglement experiments;
An experimental test of all theories with predictive power beyond quantum theory – May 2011
Excerpt: More precisely, we perform various measurements (conscious observations) on distant entangled photons, and, under the assumption that these measurements (conscious observations) are freely chosen (free will), we give a upper bound on how well any alternative theory could predict their outcomes.,,,,
Hence, we can immediately refute any already considered or yet-to-be-proposed alternative model with more predictive power than this (Quantum Theory).
Can quantum theory be improved? - July 23, 2012
Excerpt: However, in the new paper, the physicists have experimentally demonstrated that there cannot exist any alternative theory that increases the predictive probability of quantum theory by more than 0.165, with the only assumption being that measurement (conscious observation) parameters can be chosen independently (free will assumption) of the other parameters of the theory.,,,
,, the experimental results provide the tightest constraints yet on alternatives to quantum theory.,,,
Now this is completely unheard of in science as far as I know. i.e. That a mathematical description of reality would advance to the point that one can actually perform a experiment showing that your current mathematical theory will not be exceeded in predictive power by another future mathematical theory is simply unprecedented in the history of science! It is, in my unsolicited opinion, a very significant milestone in the history of science! Moreover, the belief that 'maths just is' is simply completely demolished by the fact that our best mathematical description of reality is absolutely dependent on the starting assumptions of conscious observation and free will. Moreover, since our best mathematical description of reality requires conscious observation and free will as starting assumptions, then this necessarily implies that consciousness and free will precede the mathematical equation.
Of note: since our free will choices figure so prominently in how reality is actually found to be constructed in our understanding of quantum mechanics, I think a Christian perspective on just how important our choices are in this temporal life, in regards to our eternal destiny, is very fitting:
Is God Good? (Free will and the problem of evil) - video
i.e. God gives us a ‘free will’ because without true free will it is impossible to have true love. i.e. How much love would you feel if you made a robot to tell you how much it loves you?
Music and Verse:
Third Day – Trust In Jesus
I call heaven and earth to witness against you today, that I have set before you life and death, blessing and curse. Therefore choose life, that you and your offspring may live,
Ravi Zacharias - How To Measure Your Choices - video
Dr Zacharias states, you must measure your choices by the measure of:
A few more notes on the ‘spirituality of math’:
It is interesting to note that ‘higher dimensional’ mathematics had to be developed before Einstein could elucidate General Relativity, or even before Quantum Mechanics could be elucidated;
The Mathematics Of Higher Dimensionality – Gauss and Riemann – video
Centrality of Earth Within The 4-Dimensional Space-Time of General Relativity – video
Dr. Quantum – Double Slit Experiment & Entanglement – video
The Unreasonable Effectiveness of Mathematics in the Natural Sciences – Eugene Wigner – 1960
Excerpt: We now have, in physics, two theories of great power and interest: the theory of quantum phenomena and the theory of relativity.,,, The two theories operate with different mathematical concepts: the four dimensional Riemann space and the infinite dimensional Hilbert space,
When one looks at the 4-D space time of relativity, and the centrality of conscious observation in quantum mechanics, a very interesting ‘anomaly’ pops out:
The Galileo Affair and the true “Center of the Universe”
Excerpt: I find it extremely interesting, and strange, that quantum mechanics tells us that instantaneous quantum wave collapse to its ‘uncertain’ 3-D state is centered on each individual observer in the universe, whereas, 4-D space-time cosmology (General Relativity) tells us each 3-D point in the universe is central to the expansion of the universe. These findings of modern science are pretty much exactly what we would expect to see if this universe were indeed created, and sustained, from a higher dimension by a omniscient, omnipotent, omnipresent, eternal Being who knows everything that is happening everywhere in the universe at the same time. These findings certainly seem to go to the very heart of the age old question asked of many parents by their children, “How can God hear everybody’s prayers at the same time?”,,, i.e. Why should the expansion of the universe, or the quantum wave collapse of the entire universe, even care that you or I, or anyone else, should exist? Only Theism offers a rational explanation as to why you or I, or anyone else, should have such undeserved significance in such a vast universe:
The LORD looks from heaven; He sees all the sons of men. From the place of His dwelling He looks on all the inhabitants of the earth; He fashions their hearts individually; He considers all their works.
The following is another very ‘spiritual’ finding from mathematics:
The Scale of The Universe – Part 2 – interactive graph (recently updated in 2012 with cool features)
The preceding interactive graph points out that the smallest scale visible to the human eye (as well as a human egg) is at 10^-4 meters, which ‘just so happens’ to be directly in the exponential center of all possible sizes of our physical reality (not just ‘nearly’ in the exponential center!). i.e. 10^-4 is, exponentially, right in the middle of 10^-35 meters, which is the smallest possible unit of length, which is Planck length, and 10^27 meters, which is the largest possible unit of ‘observable’ length since space-time was created in the Big Bang, which is the diameter of the universe. This is very interesting for, as far as I can tell, the limits to human vision (as well as the size of the human egg) could have, theoretically, been at very different positions than directly in the exponential middle;
Here is another finding from mathematics that has very strong ‘spiritual’ implications:
There is also a very mysterious ‘higher dimensional’ component found in life:
The predominance of quarter-power (4-D) scaling in biology
Excerpt: Many fundamental characteristics of organisms scale
with body size as power laws of the form:
Y = Yo M^b,
where Y is some characteristic such as metabolic rate, stride length or life span, Yo is a normalization constant, M is body mass and b is the allometric scaling exponent.
A longstanding puzzle in biology is why the exponent b is usually some simple multiple of 1/4 (4-Dimensional scaling) rather than a multiple of 1/3, as would be expected from Euclidean (3-Dimensional) scaling.
“Although living things occupy a three-dimensional space, their internal physiology and anatomy operate as if they were four-dimensional. Quarter-power scaling laws are perhaps as universal and as uniquely biological as the biochemical pathways of metabolism, the structure and function of the genetic code and the process of natural selection.,,, The conclusion here is inescapable, that the driving force for these invariant scaling laws cannot have been natural selection.”
Jerry Fodor and Massimo Piatelli-Palmarini, What Darwin Got Wrong (London: Profile Books, 2010), p. 78-79
Though Jerry Fodor and Massimo Piatelli-Palmarini rightly find it inexplicable for ‘random’ Natural Selection to be the rational explanation for the invariant scaling of the physiology, and anatomy, of living things to four-dimensional parameters, they do not seem to fully realize the implications this ‘four dimensional scaling’ of living things presents. This 4-D scaling is something we should rightly expect from a Intelligent Design perspective. This is because Intelligent Design holds that ‘higher dimensional transcendent information’ is more foundational to life, and even to the universe itself, than either matter or energy are. This higher dimensional ‘expectation’ for life, from a Intelligent Design perspective, is directly opposed to the expectation of the Darwinian framework, which holds that information, and indeed even the essence of life itself, is merely an ‘emergent’ property of the 3-D material realm.
Earth’s crammed with heaven,
And every common bush afire with God;
But only he who sees, takes off his shoes,
The rest sit round it and pluck blackberries.
– Elizabeth Barrett Browning
Music and verse:
YOU ARE GOD ALONE, Philips, Craig and Dean
Thou art worthy, O Lord, to receive glory and honour and power: for thou hast created all things, and for thy pleasure they are and were created.
Here is the entire video:
Is Faith in God Reasonable? FULL DEBATE with William Lane Craig and Alex Rosenberg - video
Let There Be Light - Intelligent Design