God and mathematics – Why does mathematics work?

_{Denyse O'Leary

August 23, 2020

Intelligent Design, Mathematics, theism}

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From William Lane Craig’s Reasonable Faith. With special apologies, of course, to the war on math people:

Think about it… Mathematical entities like numbers, sets, and equations are non-physical and abstract. They can’t cause anything. Yet, for some reason, the physical universe operates…mathematically.
As Galileo put it, “The book of nature is written in the language of mathematics.”
“Scientists do not use mathematics merely as a convenient way of organising the data. They believe that mathematical relationships reflect real aspects of the physical world. Science relies on the assumption that we live in an ordered Universe that is subject to precise mathematical laws. Thus, the laws of physics, … are all expressed as mathematical equations.” – Paul Davies
So, how do we explain the astonishing applicability of math to the physical world?

Those who think math doesn’t apply have probably bought into the oppressive idea that 2 + 2 = 4 And all social ills can be traced to that awful myth. 😉

Hat tip: Philip Cunningham

Comments

The reductive materialism that Darwinian evolution is based upon is simply a non-starter for ever explaining, number 1, the existence of the mathematical laws of the universe,,
“There cannot be, in principle, a naturalistic bottom-up explanation for immutable physical laws — which are themselves an ‘expression’ of top-down causation. A bottom-up explanation, from the level of e.g. bosons, should be expected to give rise to innumerable different ever-changing laws. By analogy, particles give rise to innumerable different conglomerations. Moreover a bottom-up process from bosons to physical laws is in need of constraints (laws) in order to produce a limited set of universal laws. Paul Davies: “Physical processes, however violent or complex, are thought to have absolutely no effect on the laws. There is thus a curious asymmetry: physical processes depend on laws but the laws do not depend on physical processes. Although this statement cannot be proved, it is widely accepted.” Saying that laws do not depend on physical processes, is another way of saying that laws cannot be explained by physical processes.” – Origenes
,,, and the reductive materialism that Darwinian evolution is based upon is also a non-starter for ever explaining, number 2, why man, as a supposedly purely material being, has the capacity within himself to comprehend this immaterial realm of mathematics (and logic).
Naturalism and Self-Refutation – Michael Egnor – January 31, 2018 Excerpt: Mathematics is certainly something we do. Is mathematics “included in the space-time continuum [with] basic elements … described by physics”?,,, What is the physics behind the Pythagorean theorem? After all, no actual triangle is perfect, and thus no actual triangle in nature has sides such that the Pythagorean theorem holds. There is no real triangle in which the sum of the squares of the sides exactly equals the square of the hypotenuse. That holds true for all of geometry. Geometry is about concepts, not about anything in the natural world or about anything that can be described by physics. What is the “physics” of the fact that the area of a circle is pi multiplied by the square of the radius? And of course what is natural and physical about imaginary numbers, infinite series, irrational numbers, and the mathematics of more than three spatial dimensions? Mathematics is entirely about concepts, which have no precise instantiation in nature,,, Furthermore, the very framework of Clark’s argument — logic — is neither material nor natural. Logic, after all, doesn’t exist “in the space-time continuum” and isn’t described by physics. What is the location of modus ponens? How much does Gödel’s incompleteness theorem weigh? What is the physics of non-contradiction? How many millimeters long is Clark’s argument for naturalism? Ironically the very logic that Clark employs to argue for naturalism is outside of any naturalistic frame. The strength of Clark’s defense of naturalism is that it is an attempt to present naturalism’s tenets clearly and logically. That is its weakness as well, because it exposes naturalism to scrutiny, and naturalism cannot withstand even minimal scrutiny. Even to define naturalism is to refute it. https://evolutionnews.org/2018/01/naturalism-and-self-refutation/
The fact that man himself can comprehend this transcendent, beyond space and time, 'Platonic world' of mathematics, offers fairly compelling evidence, in and of itself, that man must also possess an immaterial mind and soul. As Charles Darwin’s contemporary, Alfred Russel Wallace himself stated, “Nothing in evolution can account for the soul of man. The difference between man and the other animals is unbridgeable. Mathematics is alone sufficient to prove in man the possession of a faculty unexistent in other creatures. Then you have music and the artistic faculty. No, the soul was a separate creation.”
“Nothing in evolution can account for the soul of man. The difference between man and the other animals is unbridgeable. Mathematics is alone sufficient to prove in man the possession of a faculty unexistent in other creatures. Then you have music and the artistic faculty. No, the soul was a separate creation.” Alfred Russel Wallace – 1910 https://evolutionnews.org/2010/08/alfred_russel_wallace_co-disco/
And as Michael Egnor stated, "Human rationality is different because it is immaterial. Contemplation of universals cannot have material instantiation, because universals themselves are not material and cannot be instantiated in matter.,, We are more different from apes than apes are from viruses. Our difference is a metaphysical chasm,,,"
The Fundamental Difference Between Humans and Nonhuman Animals - Michael Egnor - November 5, 2015 Excerpt: Human beings have mental powers that include the material mental powers of animals but in addition entail a profoundly different kind of thinking. Human beings think abstractly, and nonhuman animals do not. Human beings have the power to contemplate universals, which are concepts that have no material instantiation. Human beings think about mathematics, literature, art, language, justice, mercy, and an endless library of abstract concepts. Human beings are rational animals. Human rationality is not merely a highly evolved kind of animal perception. Human rationality is qualitatively different — ontologically different — from animal perception. Human rationality is different because it is immaterial. Contemplation of universals cannot have material instantiation, because universals themselves are not material and cannot be instantiated in matter. ,,, It is in our ability to think abstractly that we differ from apes. It is a radical difference — an immeasurable qualitative difference, not a quantitative difference. We are more different from apes than apes are from viruses. Our difference is a metaphysical chasm,,, https://evolutionnews.org/2015/11/the_fundamental_2/
Verse:
Mark 8:37 Is anything worth more than your soul?
Moreover, it is not only that Darwinian materialists lack an explanation for why the universe is described by mathematics, or for why we have the capacity within ourselves to comprehend this immaterial world of mathematics, it is that the materialism that undergirds Darwinian thought actually denies the existence of anything beyond the material realm. i.e. Darwinian materialism actually denies the existence of immaterial mathematics!
What Does It Mean to Say That Science & Religion Conflict? - M. Anthony Mills - April 16, 2018 Excerpt: In fact, more problematic for the materialist than the non-existence of persons is the existence of mathematics. Why? Although a committed materialist might be perfectly willing to accept that you do not really exist, he will have a harder time accepting that numbers do not exist. The trouble is that numbers — along with other mathematical entities such as classes, sets, and functions — are indispensable for modern science. And yet — here’s the rub — these “abstract objects” are not material. Thus, one cannot take science as the only sure guide to reality and at the same time discount disbelief in all immaterial realities. https://www.realclearreligion.org/articles/2018/04/16/what_does_it_mean_to_say_that_science_and_religion_conflict.html
And seeing that mathematics undergirds all of science, engineering, and technology, and seeing that the reductive materialism that Darwinian evolution rests upon actually denies the very existence of immaterial mathematics, then, of course, Darwinian materialism immediately disqualifies itself from ever being considered a truly 'scientific' worldview in the first place. Moreover, Darwinian atheists may not like it one bit, but our use of immaterial mathematics to describe the laws of the universe is VERY harmonious to a Theistic view of reality. As Paul Davies explained,
“People take it for granted that the physical world is both ordered and intelligible. The underlying order in nature-the laws of physics-are simply accepted as given, as brute facts. Nobody asks where they came from; at least they do not do so in polite company. However, even the most atheistic scientist accepts as an act of faith that the universe is not absurd, that there is a rational basis to physical existence manifested as law-like order in nature that is at least partly comprehensible to us. So science can proceed only if the scientist adopts an essentially theological worldview.” —Paul Davies (cited in, The Historic Alliance of Christianity and Science)
And as David Berlinski stated, "There is no argument against religion that is not also an argument against mathematics. Mathematicians are capable of grasping a world of objects that lies beyond space and time…."
An Interview with David Berlinski - Jonathan Witt Berlinski: There is no argument against religion that is not also an argument against mathematics. Mathematicians are capable of grasping a world of objects that lies beyond space and time…. Interviewer:… Come again(?) … Berlinski: No need to come again: I got to where I was going the first time. The number four, after all, did not come into existence at a particular time, and it is not going to go out of existence at another time. It is neither here nor there. Nonetheless we are in some sense able to grasp the number by a faculty of our minds. Mathematical intuition is utterly mysterious. So for that matter is the fact that mathematical objects such as a Lie Group or a differentiable manifold have the power to interact with elementary particles or accelerating forces. But these are precisely the claims that theologians have always made as well – that human beings are capable by an exercise of their devotional abilities to come to some understanding of the deity; and the deity, although beyond space and time, is capable of interacting with material objects. http://tofspot.blogspot.com/2013/10/found-upon-web-and-reprinted-here.html
Verse and quotes:
John 1:1 “In the beginning was the Word, and the Word was with God, and the Word was God” What is the Logos? Logos is a Greek word literally translated as “word, speech, or utterance.” However, in Greek philosophy, Logos refers to divine reason or the power that puts sense into the world making order instead of chaos.,,, In the Gospel of John, John writes “In the beginning was the Word (Logos), and the Word was with God, and the Word was God” (John 1:1). John appealed to his readers by saying in essence, “You’ve been thinking, talking, and writing about the Word (divine reason) for centuries and now I will tell you who He is.” https://www.compellingtruth.org/what-is-the-Logos.html “Atheists can give no reason why they should value reason, and Christians can show how anyone who believes in reason must also believe in God.” Cogito; Ergo Deus Est by Charles Edward White Philosophy Still Lives Because God Isn't Dead
bornagain77_{August 24, 2020
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JVL, You might want to read some books on quantum mechanics and general relativity. I'm not going to write one for you in a reply. LOL -QQuerius_{August 24, 2020
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Seversky
The Universe works in certain ways and we have invented a very flexible and powerful language which we can use to model how it works.
Under evo-materialism, we have invented nothing. Certain brain structures/ neurons have been "selected" because MAGICALLY (materialism's key word) they "produced mathematics". Or even worse, they are "spandrels" (not even f***ng selected, just "free-riders") that MAGICALLY (materialism's key word) "create maths". How the hell can a brain "create mathematics"? What neurochemicals are "math-creators"? Does the brain "secrete" 2+2? Does an accountant (homunculi) live inside our heads? And more important: does he/she get paid a fair wage? Materialism= magical non-sense.Truthfreedom_{August 24, 2020
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JVL
In general I agree with Seversky...
Asinus asinum fricat.Truthfreedom_{August 24, 2020
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Querius: The Logos forms, ties together, and maintains two currently incompatible mathematical descriptions of reality. What two descriptions are those? The problem with JVL’s assertion is that “inanimate physical objects” ARE fundamentally mathematical probabilities according to quantum mechanics! Show me the work behind this assertion. But to have probabilities, including quantum fluctuations, one needs Time. No Time equals no probabilities or fluctuations. According to Einstein, Space and Time are part of a single, unified manifold. Once you have space-time, you can now have mathematical probabilities that manifest themselves as “quantum foam” or mass-energy. For example?JVL_{August 24, 2020
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Kushanto: What other suggestions do you have, considering that nothing comes from absolutely nothing? I don't have any other suggestions; I am merely pointing out that we don't know. AND who ever said that nothing came from nothing?JVL_{August 24, 2020
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ET: It still applies. Why? On what grounds? Nice quote-mine. Are you 3 years old? I offer my opinion and you accuse me of quote mining? Neither mathematics nor living organisms would exist if the universe wasn’t intelligently designed using mathematics. Mathematics is directly linked to the laws that govern the universe. Again, this is pure assertion with no hard evidence to back it up. I appreciate that this is your opinion but it is not fact. We don't know.JVL_{August 24, 2020
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Bornagain77 @9 and 10, Great answer! The Logos forms, ties together, and maintains two currently incompatible mathematical descriptions of reality. KF @12 articulates this concept excellently in mathematical terms! Regarding KF @14, Let me agree that the mathematics identifies a relationship between components similar to a state diagram. In two senses, it's not causal. In other words, in one sense the relationships could not have caused themselves, and in the other sense, a state diagram is static without a push. But . . .
JVL: I was talking about mathematics not inanimate physical objects.
The problem with JVL's assertion is that "inanimate physical objects" ARE fundamentally mathematical probabilities according to quantum mechanics! But to have probabilities, including quantum fluctuations, one needs Time. No Time equals no probabilities or fluctuations. According to Einstein, Space and Time are part of a single, unified manifold. Once you have space-time, you can now have mathematical probabilities that manifest themselves as "quantum foam" or mass-energy. Information and logic govern these. Reminds me of Genesis 1:1-5 and John 1:1-3. -QQuerius_{August 24, 2020
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JVL@15 "We don’t know that. Just asserting it’s true proves nothing. And don’t get snippy about the word prove, in mathematics you can prove things" What other suggestions do you have, considering that nothing comes from absolutely nothing?Kushanto_{August 24, 2020
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JVL:
I was talking about mathematics not inanimate physical objects.
It still applies.
We don’t know that.
Nice quote-mine. Are you 3 years old? Neither mathematics nor living organisms would exist if the universe wasn't intelligently designed using mathematics. Mathematics is directly linked to the laws that govern the universe.ET_{August 24, 2020
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ES, though logic of structure and quantity is not actively causal, logical impossibility of being due to contradictory core characteristics -- square circles -- and necessity of being tracing to say that a distinct possible world is just that, distinct and possible, will constrain what is or can be. Thus we come to structures and quantities inherent in distinct possible worlds, thus necessary structured frameworks that define the core of mathematics. See the paper linked at 1. KFkairosfocus_{August 24, 2020
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Seversky: "the map is not the territory so is a model not the same as what is being modeled." Exactly. That is why mathematics is not causative. That is why saying, as Hawking did, that gravitation is a result of the M-theory is nonsensical. That is why attributing the causative power to the the laws of nature is nonsensical. The laws of nature are nothing but a description of natural regularities. The laws of nature cannot cause anything to happen. As Lennox aptly said, the laws of economics did not put a single penny into my bank account. Nature had a beginning (which actually seems to be a scientific consensus today). However, its beginning is necessarily super-natural, otherwise there is a contradiction. Science can cope with providing descriptions of how the already existing matter/energy behave over time but, necessarily, being applied to the question of how it all came to be, science is on fool's errand and reduces to sci-fi.EugeneS_{August 24, 2020
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ET: What JVL is saying is like saying whether or not Stonehenge was invented or discovered is philosophical. I was talking about mathematics not inanimate physical objects. And we know that is total nonsense. Mathematics work only because the universe was Intelligently Designed using mathematics. We don't know that. Just asserting it's true proves nothing. And don't get snippy about the word prove, in mathematics you can prove things.JVL_{August 24, 2020
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What JVL is saying is like saying whether or not Stonehenge was invented or discovered is philosophical. And we know that is total nonsense. Mathematics work only because the universe was Intelligently Designed using mathematics.ET_{August 24, 2020
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Belfast: Mathematical relationships existed long before they were discovered. Pythagoras described the relationship between the sides of a right angled triangle on a flat plain – he didn’t invent it. Umm, I agreed with that. Did you actually read what I wrote? The comparison given with a map is is simplistic to the point of craziness – produce an equation that describes, say, a street scene, make up your own variables. I admitted the analogy wasn't perfect but a map and an equation have some similar characteristics. In some sense vector graphics are groups of equations describing an image. So, if you took a photo of a street scene, converted it to vector graphics you would get a series of equations.JVL_{August 24, 2020
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Seversky & JVL, you would profit from reading the paper JB links at 1. Mathematics has a substance embedded in reality expressed in possible worlds including this actual one. There is a study that in key parts explores and discovers patterns, features, entities and structures, antecedent to axiomatisations. Axiomatisations on key points, are accountable to the body of facts [eg || + ||| --> ||||| ] and insofar as they reflect the structure well, allow us to further explore. Where, mathematics as study can then be seen as the study of the logic of structure and quantity. Axiom systems, which perhaps capture a too central part of modern conceptions are logic world constructs. Insofar as they reflect necessary being aspects of structure and quantity they apply to any possible world. In other aspects, there is more flexibility and they may or may not reflect our world. Here we blend over into scientific laws, theories and models . . . including those useful as they are good enough for practical work. KFkairosfocus_{August 24, 2020
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For the Christian this theoretical finding from our very best theories in science, (i.e. that something very powerful must be ‘outside the universe’ that is holding this universe together), should not be all that surprising to find out. Christianity, a couple of millennium before the 'infinite' conflict between the General Relativity and Quantum Mechanics was even known about, predicted that Christ is before all things, and 'in him all things hold together,,,'
Colossians 1:17 He is before all things, and in him all things hold together.
Dr. William Dembski in this following comment, although he was not directly addressing the 'infinite' conflict between General Relativity and Quantum Mechanics, offers this insight into what the ‘unification’ of infinite God with finite man might look like mathematically:
The End Of Christianity – Finding a Good God in an Evil World – Pg.31 William Dembski PhDs. Mathematics and Theology Excerpt: “In mathematics there are two ways to go to infinity. One is to grow large without measure. The other is to form a fraction in which the denominator goes to zero. The Cross is a path of humility in which the infinite God becomes finite and then contracts to zero, only to resurrect and thereby unite a finite humanity within a newfound infinity.” http://www.designinference.com/documents/2009.05.end_of_xty.pdf Philippians 2:8-9 And being found in appearance as a man, He humbled Himself and became obedient to the point of death, even the death of the cross. Therefore God also has highly exalted Him and given Him the name which is above every name,
Moreover, if we rightly allow the Agent causality of God ‘back’ into physics, as the Christian founders of modern science originally envisioned,,,, (Isaac Newton, Michael Faraday, James Clerk Maxwell, and Max Planck, to name a few of the Christian founders),,, and as quantum mechanics itself now empirically demands (with the closing of the free will loophole by Anton Zeilinger and company), if we rightly allow the Agent causality of God ‘back’ into physics, then that provides us with a very plausible resolution for the much sought after ‘theory of everything’ in that Christ’s resurrection from the dead provides an ’empirically backed’ reconciliation, via the Shroud of Turin, between quantum mechanics and general relativity into the much sought after ‘Theory of Everything”. Here are a few posts where I lay out and defend some of the evidence for that claim:
November 2019 – despite the fact that virtually everyone, including the vast majority of Christians, hold that the Copernican Principle is unquestionably true, the fact of the matter is that the Copernican Principle is now empirically shown, (via quantum mechanics and general relativity, etc..), to be a false assumption. https://uncommondescent.com/intelligent-design/so-then-maybe-we-are-privileged-observers/#comment-688855 (February 19, 2019) To support Isabel Piczek’s claim that the Shroud of Turin does indeed reveal a true ‘event horizon’, the following study states that ‘The bottom part of the cloth (containing the dorsal image) would have born all the weight of the man’s supine body, yet the dorsal image is not encoded with a greater amount of intensity than the frontal image.’,,, Moreover, besides gravity being dealt with, the shroud also gives us evidence that Quantum Mechanics was dealt with. In the following paper, it was found that it was not possible to describe the image formation on the Shroud in classical terms but they found it necessary to describe the formation of the image on the Shroud in discrete quantum terms. https://uncommondescent.com/intelligent-design/experiment-quantum-particles-can-violate-the-mathematical-pigeonhole-principle/#comment-673178 The evidence for the Shroud’s authenticity keeps growing. (Timeline of facts) – November 08, 2019 What Is the Shroud of Turin? Facts & History Everyone Should Know – Myra Adams and Russ Breault https://www.christianity.com/wiki/jesus-christ/what-is-the-shroud-of-turin.html
To give us a small glimpse of the power that was involved in Christ’s resurrection from the dead, the following recent article found that, ”it would take 34 Thousand Billion Watts of VUV radiations to make the image on the shroud. This output of electromagnetic energy remains beyond human technology.”
Astonishing discovery at Christ’s tomb supports Turin Shroud – NOV 26TH 2016 Excerpt: The first attempts made to reproduce the face on the Shroud by radiation, used a CO2 laser which produced an image on a linen fabric that is similar at a macroscopic level. However, microscopic analysis showed a coloring that is too deep and many charred linen threads, features that are incompatible with the Shroud image. Instead, the results of ENEA “show that a short and intense burst of VUV directional radiation can color a linen cloth so as to reproduce many of the peculiar characteristics of the body image on the Shroud of Turin, including shades of color, the surface color of the fibrils of the outer linen fabric, and the absence of fluorescence”. ‘However, Enea scientists warn, “it should be noted that the total power of VUV radiations required to instantly color the surface of linen that corresponds to a human of average height, body surface area equal to = 2000 MW/cm2 17000 cm2 = 34 thousand billion watts makes it impractical today to reproduce the entire Shroud image using a single laser excimer, since this power cannot be produced by any VUV light source built to date (the most powerful available on the market come to several billion watts )”. Comment The ENEA study of the Holy Shroud of Turin concluded that it would take 34 Thousand Billion Watts of VUV radiations to make the image on the shroud. This output of electromagnetic energy remains beyond human technology. http://westvirginianews.blogspot.com/2011/12/new-study-claims-shroud-of-turin-is.html
Verse:
Colossians 1:15-20 The Son is the image of the invisible God, the firstborn over all creation. For in him all things were created: things in heaven and on earth, visible and invisible, whether thrones or powers or rulers or authorities; all things have been created through him and for him. He is before all things, and in him all things hold together. And he is the head of the body, the church; he is the beginning and the firstborn from among the dead, so that in everything he might have the supremacy. For God was pleased to have all his fullness dwell in him, and through him to reconcile to himself all things, whether things on earth or things in heaven, by making peace through his blood, shed on the cross.
Thus in conclusion, and although much more could be said about this topic, the main impasse for modern day physicists today in finding the ‘theory of everything’, apparently, seems to be that modern day physicists have, basically, completely forgotten the philosophical, i.e. Christian, roots that gave rise to modern science in the first place, i.e. That the universe, and math itself, are both the result of the ‘thoughts of God’. And have regressed back into ancient Greek rationalism in which math served as a rival to God rather than a path to Him. In short, modern day theoretical physicists have apparently, for the most part, forgotten the philosophical presuppositions that enabled the Christian founders of modern science to make their breakthrough into modern science in. the first place.
“O, Almighty God, I am thinking Thy thoughts after Thee!” Johannes Kepler – 1619 – stated shortly after he discovered the laws of planetary motion.
bornagain77_{August 24, 2020
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Bottom line, due to Godel's incompleteness theorem(s), even if a single overarching mathematical theory happened to be found that described the universe (instead of the two that we currently have, (i.e. General Relativity and Quantum Mechanics), we still would find it necessary to postulate God for why that particular equation, out of an infinity of other possible mathematical descriptions, happened to describe the universe:
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. http://www.firstthings.com/article/2010/08/the-god-of-the-mathematicians BRUCE GORDON: Hawking’s irrational arguments – October 2010 Excerpt: ,,,The physical universe is causally incomplete and therefore neither self-originating nor self-sustaining. The world of space, time, matter and energy is dependent on a reality that transcends space, time, matter and energy. This transcendent reality cannot merely be a Platonic realm of mathematical descriptions, for such things are causally inert abstract entities that do not affect the material world,,, Rather, the transcendent reality on which our universe depends must be something that can exhibit agency – a mind that can choose among the infinite variety of mathematical descriptions and bring into existence a reality that corresponds to a consistent subset of them. This is what “breathes fire into the equations and makes a universe for them to describe.” Anything else invokes random miracles as an explanatory principle and spells the end of scientific rationality. http://www.washingtontimes.com/news/2010/oct/1/hawking-irrational-arguments/
Yet we do not have just one mathematical 'theory of everything' that describes the universe. We have two theories, General Relativity and Quantum Mechanics, that simply refuse to unified into a single overarching 'theory of everything'. In fact, there is an infinite mathematical divide that separates the two theories. As Sera Cremonini states, ” the quantum version of Einstein’s general relativity is “nonrenormalizable.”,,, “The problem with a quantum version of general relativity is that the calculations that would describe interactions of very energetic gravitons — the quantized units of gravity — would have infinitely many infinite terms. You would need to add infinitely many counterterms in a never-ending process. Renormalization would fail.,,,”
Why Gravity Is Not Like the Other Forces We asked four physicists why gravity stands out among the forces of nature. We got four different answers. Excerpt: the quantum version of Einstein’s general relativity is “nonrenormalizable.”,,, In quantum theories, infinite terms appear when you try to calculate how very energetic particles scatter off each other and interact. In theories that are renormalizable — which include the theories describing all the forces of nature other than gravity — we can remove these infinities in a rigorous way by appropriately adding other quantities that effectively cancel them, so-called counterterms. This renormalization process leads to physically sensible answers that agree with experiments to a very high degree of accuracy. The problem with a quantum version of general relativity is that the calculations that would describe interactions of very energetic gravitons — the quantized units of gravity — would have infinitely many infinite terms. You would need to add infinitely many counterterms in a never-ending process. Renormalization would fail.,,, Sera Cremonini – theoretical physicist – Lehigh University https://www.quantamagazine.org/why-gravity-is-not-like-the-other-forces-20200615/
Likewise, Professor Jeremy Bernstein states the situation as such, “there remains an irremediable difficulty. Every order reveals new types of infinities, and no finite number of renormalizations renders all the terms in the series finite.”
Quantum Leaps – Jeremy Bernstein – October 19, 2018 Excerpt: Divergent series notwithstanding, quantum electrodynamics yielded results of remarkable accuracy. Consider the magnetic moment of the electron. This calculation, which has been calculated up to the fifth order in ?, agrees with experiment to ten parts in a billion. If one continued the calculation to higher and higher orders, at some point the series would begin to break down. There is no sign of that as yet. Why not carry out a similar program for gravitation? One can readily write down the Feynman graphs that represent the terms in the expansion. Yet there remains an irremediable difficulty. Every order reveals new types of infinities, and no finite number of renormalizations renders all the terms in the series finite. The theory is not renormalizable. https://inference-review.com/article/quantum-leaps Jeremy Bernstein is professor emeritus of physics at the Stevens Institute of Technology.
Moreover, when theorists try to combine the two theories, then the resulting theory predicts that spacetime, atoms, and even the universe itself should all be literally blown apart. Here are a few references that get this point across.
“There are serious problems with the traditional view that the world is a space-time continuum. Quantum field theory and general relativity contradict each other. The notion of space-time breaks down at very small distances, because extremely massive quantum fluctuations (virtual particle/antiparticle pairs) should provoke black holes and space-time should be torn apart, which doesn’t actually happen.” – Gregory J. Chaitin , Francisco A. Doria, and Newton C. a. Da Costa – Goedel’s Way: Exploits into an Undecidable World “In order for quantum mechanics and relativity theory to be internally self-consistent [Seeking consistency between quantum mechanics and relativity theory is the major task theoretical physicists have been grappling with since quantum mechanics emerged], the physical vacuum has to contain 10^94 grams equivalent of energy per cubic centimeter. What that means is, if you take just a single hydrogen atom, which is one proton and one electron and all the rest of the atom is ‘empty space,’ if you take just that volume of empty space, … you find that you end up with a trillion times as much vacuum energy as all the electromagnetic energy in all the planets, all the stars, and all the cosmic dust in a sphere of radius 15 billion light-years.” To summarize, the subtle energy in the vacuum space of a single hydrogen atom is as great as all the electromagnetic energy found in everything within 15 billion light-years of our space-time cosmos.” ,,, Dr. William Tiller – Human Intention Cosmic coincidence spotted – Philip Ball – 2008 Excerpt: One interpretation of dark energy is that it results from the energy of empty space, called vacuum energy. The laws of quantum physics imply that empty space is not empty at all, but filled with particles popping in and out of existence. This particle ‘fizz’ should push objects apart, just as dark energy seems to require. But the theoretical value of this energy is immense — so huge that it should blow atoms apart, rather than just causing the Universe to accelerate. Physicists think that some unknown force nearly perfectly cancels out the vacuum energy, leaving only the amount seen as dark energy to push things apart. This cancellation is imperfect to an absurdly fine margin: the unknown ‘energy’ differs from the vacuum energy by just one part in 10^122. It seems incredible that any physical mechanism could be so finely poised as to reduce the vacuum energy to within a whisker of zero, but it seems to be so. http://www.nature.com/news/2008/080219/full/news.2008.610.html The 2 most dangerous numbers in the universe are threatening the end of physics – Jessica Orwig – Jan. 14, 2016 Excerpt: Dangerous No. 2: The strength of dark energy ,,, you should be able to sum up all the energy of empty space to get a value representing the strength of dark energy. And although theoretical physicists have done so, there’s one gigantic problem with their answer: “Dark energy should be 10^120 times stronger than the value we observe from astronomy,” Cliff said. “This is a number so mind-boggling huge that it’s impossible to get your head around … this number is bigger than any number in astronomy — it’s a thousand-trillion-trillion-trillion times bigger than the number of atoms in the universe. That’s a pretty bad prediction.” On the bright side, we’re lucky that dark energy is smaller than theorists predict. If it followed our theoretical models, then the repulsive force of dark energy would be so huge that it would literally rip our universe apart. The fundamental forces that bind atoms together would be powerless against it and nothing could ever form — galaxies, stars, planets, and life as we know it would not exist. http://finance.yahoo.com/news/two-most-dangerous-numbers-universe-194557366.html
And yet, despite both theories contradicting each other to the point of literally blowing the universe apart, the fact remains that quantum mechanics and general relativity are both tested to extreme levels of precision, (in fact, both general relativity and quantum mechanics are consider to be our most successful theories ever in the history of science),
The Most Precisely Tested Theory in the History of Science – May 5, 2011 Excerpt: So, which of the two (general relativity or QED) is The Most Precisely Tested Theory in the History of Science? It’s a little tough to quantify a title like that, but I think relativity can claim to have tested the smallest effects. Things like the aluminum ion clock experiments showing shifts in the rate of a clock set moving at a few m/s, or raised by a foot, measure relativistic shifts of a few parts in 10^16. That is, if one clock ticks 10,000,000,000,000,000 times, the other ticks 9,999,999,999,999,999 times. That’s an impressively tiny effect, but the measured value is in good agreement with the predictions of relativity. In the end, though, I have to give the nod to QED, because while the absolute effects in relativity may be smaller, the precision of the measurements in QED is more impressive. Experimental tests of relativity measure tiny shifts, but to only a few decimal places. Experimental tests of QED measure small shifts, but to an absurd number of decimal places. The most impressive of these is the “anomalous magnetic moment of the electron,” expressed is terms of a number g whose best measured value is: g/2 = 1.001 159 652 180 73 (28) Depending on how you want to count it, that’s either 11 or 14 digits of precision (the value you would expect without QED is exactly 1, so in some sense, the shift really starts with the first non-zero decimal place), which is just incredible. And QED correctly predicts all those decimal places (at least to within the measurement uncertainty, given by the two digits in parentheses at the end of that). http://scienceblogs.com/principles/2011/05/05/the-most-precisely-tested-theo/
And since quantum mechanics and general relativity are both tested to such an extreme level of precision, (and we can thus have a very high level of confidence that both theories are, in fact, true mathematical descriptions of reality), and since Godel’s incompleteness theorem itself requires something to be ‘outside the circle’ of mathematics (Hawking),," then it is fairly safe to assume that there must be something very powerful that must be holding the universe together in order to keep it from blowing itself apart. ,,,bornagain77_{August 24, 2020
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A crucial belief that allowed the rise of modern science in medieval Christian Europe, was not only the Christian's belief that the universe itself is contingent upon the Mind of God for its existence,,,
The War against the War Between Science and Faith Revisited – July 2010 Excerpt: …If science suffered only stillbirths in ancient cultures, how did it come to its unique viable birth? The beginning of science as a fully fledged enterprise took place in relation to two important definitions of the Magisterium of the Church. The first was the definition at the Fourth Lateran Council in the year 1215, that the universe was created out of nothing at the beginning of time. The second magisterial statement was at the local level, enunciated by Bishop Stephen Tempier of Paris who, on March 7, 1277, condemned 219 Aristotelian propositions, so outlawing the deterministic and necessitarian views of creation. These statements of the teaching authority of the Church expressed an atmosphere in which faith in God had penetrated the medieval culture and given rise to philosophical consequences. The cosmos was seen as contingent in its existence and thus dependent on a divine choice which called it into being; the universe is also contingent in its nature and so God was free to create this particular form of world among an infinity of other possibilities. Thus the cosmos cannot be a necessary form of existence; and so it has to be approached by a posteriori investigation. The universe is also rational and so a coherent discourse can be made about it. Indeed the contingency and rationality of the cosmos are like two pillars supporting the Christian vision of the cosmos. http://www.scifiwright.com/2010/07/the-war-against-the-war-of-faith-versus-science/
,,, was not only the Christian's belief that the universe itself is contingent upon the Mind of God for its existence, but was also the Christian's belief that any mathematics that might describe this universe was also contingent upon the mind of God for its existence. As Edward Fesser notes in the following article, for Christian scholastic philosophers of the medieval period “Mathematical truths exhibit infinity, necessity, eternity, immutability, perfection, and immateriality because they are God’s thoughts,” whereas for ancient Greek philosophers, “mathematical objects such as numbers and geometrical figures exist not only independently of the material world, but also independently of any mind, including the divine mind.” As Dr. Feser noted, in Platonic thought, 'the mathematical realm is a rival to God rather than a path to him.'
KEEP IT SIMPLE – by Edward Feser – April 2020 Excerpt: Mathematics appears to describe a realm of entities with quasi-divine attributes. The series of natural numbers is infinite. That one and one equal two and two and two equal four could not have been otherwise. Such mathematical truths never begin being true or cease being true; they hold eternally and immutably. The lines, planes, and figures studied by the geometer have a kind of perfection that the objects of our experience lack. Mathematical objects seem immaterial and known by pure reason rather than through the senses. Given the centrality of mathematics to scientific explanation, it seems in some way to be a cause of the natural world and its order. How can the mathematical realm be so apparently godlike? The traditional answer, originating in Neoplatonic philosophy and Augustinian theology, is that our knowledge of the mathematical realm is precisely knowledge, albeit inchoate, of the divine mind. Mathematical truths exhibit infinity, necessity, eternity, immutability, perfection, and immateriality because they are God’s thoughts, and they have such explanatory power in scientific theorizing because they are part of the blueprint implemented by God in creating the world. For some thinkers in this tradition, mathematics thus provides the starting point for an argument for the existence of God qua supreme intellect. There is also a very different answer, in which the mathematical realm is a rival to God rather than a path to him. According to this view, mathematical objects such as numbers and geometrical figures exist not only independently of the material world, but also independently of any mind, including the divine mind. They occupy a “third realm” of their own, the realm famously described in Plato’s Theory of Forms. God used this third realm as a blueprint when creating the physical world, but he did not create the realm itself and it exists outside of him. This position is usually called Platonism since it is commonly thought to have been Plato’s own view, as distinct from that of his Neoplatonic followers who relocated mathematical objects and other Forms into the divine mind. (I put to one side for present purposes the question of how historically accurate this standard narrative is.) https://www.firstthings.com/article/2020/04/keep-it-simple
As Peter S Williams noted, "the Greeks held that this order is necessary and that one can therefore deduce its structure from first principles. Only biblical thought held that God created both form and matter, meaning that the world did not have to be as it is and that the details of its order can be discovered only by observation.”
Is Christianity Unscientific? - Peter S. Williams Excerpt: “Both Greek and biblical thought asserted that the world is orderly and intelligible. But the Greeks held that this order is necessary and that one can therefore deduce its structure from first principles. Only biblical thought held that God created both form and matter, meaning that the world did not have to be as it is and that the details of its order can be discovered only by observation.” http://www.bethinking.org/does-science-disprove-god/is-christianity-unscientific
And as Paul Davies noted, " What we now call the laws of physics they (the Christian founders of modern science) regarded as God's abstract creation: thoughts, so to speak, in the mind of God. So in doing science, they supposed, one might be able to glimpse the mind of God - an exhilarating and audacious claim."
"All the early scientists, like Newton, were religious in one way or another. They saw their science as a means of uncovering traces of God's handiwork in the universe. What we now call the laws of physics they regarded as God's abstract creation: thoughts, so to speak, in the mind of God. So in doing science, they supposed, one might be able to glimpse the mind of God - an exhilarating and audacious claim." - Paul Davies - quoted from an address following his award of the $1 million Templeton Prize in 1995 for progress in science and religion. http://ldolphin.org/bumbulis/
As an example of how the Christian founders of modern science viewed any mathematics that might describe this universe as being contingent upon the Mind of God, the following quote is perhaps the most direct. Shortly after discovering the mathematical laws of planetary motion, Kepler stated, "“O, Almighty God, I am thinking Thy thoughts after Thee!”
“O, Almighty God, I am thinking Thy thoughts after Thee!” – Johannes Kepler, 1619, The Harmonies of the World.
Likewise in 1687, Sir Isaac Newton, after discovering the mathematical law of universal gravitation, (which has been referred to as the first major unification in physics), stated that, “This most beautiful system of the sun, planets, and comets, could only proceed from the counsel and dominion of an intelligent and powerful Being.,,,This Being governs all things, not as the soul of the world, but as Lord over all;”
“This most beautiful system of the sun, planets, and comets, could only proceed from the counsel and dominion of an intelligent and powerful Being. And if the fixed stars are the centres of other like systems, these, being formed by the like wise counsel, must be all subject to the dominion of One; especially since the light of the fixed stars is of the same nature with the light of the sun, and from every system light passes into all the other systems: and lest the systems of the fixed stars should, by their gravity, fall on each other mutually, he hath placed those systems at immense distances one from another. This Being governs all things, not as the soul of the world, but as Lord over all; and on account of his dominion he is wont to be called Lord God pantokrator, or Universal Ruler;,,, The Supreme God is a Being eternal, infinite, absolutely perfect;,,, from his true dominion it follows that the true God is a living, intelligent, and powerful Being; and, from his other perfections, that he is supreme, or most perfect. He is eternal and infinite, omnipotent and omniscient; that is, his duration reaches from eternity to eternity; his presence from infinity to infinity; he governs all things, and knows all things that are or can be done. He is not eternity or infinity, but eternal and infinite; he is not duration or space, but he endures and is present. He endures for ever, and is every where present”: Sir Isaac Newton – Principia; 1687, GENERAL SCHOLIUM. http://gravitee.tripod.com/genschol.htm
As well, as Ian H. Hutchinson noted in the following article on Faraday and Maxwell, “Lawfulness was not, in their thinking, inert, abstract, logical necessity, or complete reducibility to Cartesian mechanism; rather, it was an expectation they attributed to the existence of a divine lawgiver.,,, For them, the coherence of nature resulted from its origin in the mind of its Creator.”
The Genius and Faith of Faraday and Maxwell – Ian H. Hutchinson – 2014 Conclusion: Lawfulness was not, in their thinking, inert, abstract, logical necessity, or complete reducibility to Cartesian mechanism; rather, it was an expectation they attributed to the existence of a divine lawgiver. These men’s insights into physics were made possible by their religious commitments. For them, the coherence of nature resulted from its origin in the mind of its Creator. http://www.thenewatlantis.com/publications/the-genius-and-faith-of-faraday-and-maxwell
As well in quantum mechanics and relativity, we find that both Eugene Wigner and Einstein are on record as to regarding it as a ‘miracle’ that mathematics is applicable to the universe. In fact, Einstein went so far as to disparage ‘professional atheists’ when he called it a miracle:
On the Rational Order of the World: a Letter to Maurice Solovine – Albert Einstein – March 30, 1952 Excerpt: “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. There lies the weakness of positivists and professional atheists who are elated because they feel that they have not only successfully rid the world of gods but “bared the miracles.” -Albert Einstein http://inters.org/Einstein-Letter-Solovine 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. http://www.dartmouth.edu/~matc/MathDrama/reading/Wigner.html
Godel's incompleteness theorem adds considerable weight to the Christian's belief that mathematics, especially any mathematics that might describe this universe, must be contingent upon the Mind of God for its existence. As Stephen Hawking himself honestly confessed, “Gödel’s incompleteness theorem (1931), proves that there are limits to what can be ascertained by mathematics. Kurt Gödel, halted the achievement of a unifying all-encompassing theory of everything in his theorem that: “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”.”
“Gödel’s incompleteness theorem (1931), proves that there are limits to what can be ascertained by mathematics. Kurt Gödel, halted the achievement of a unifying all-encompassing theory of everything in his theorem that: “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”.” - Stephen Hawking & Leonard Miodinow, The Grand Design (2010)
And as Stanley Jaki noted,
“Clearly then no scientific cosmology, which of necessity must be highly mathematical, can have its proof of consistency within itself as far as mathematics go. In absence of such consistency, all mathematical models, all theories of elementary particles, including the theory of quarks and gluons…fall inherently short of being that theory which shows in virtue of its a priori truth that the world can only be what it is and nothing else. This is true even if the theory happened to account for perfect accuracy for all phenomena of the physical world known at a particular time.” Stanley Jaki – Cosmos and Creator – 1980, pg. 49
bornagain77_{August 24, 2020
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You don’t know what you are talking about either, JVL. Mathematical relationships existed long before they were discovered. Pythagoras described the relationship between the sides of a right angled triangle on a flat plain - he didn’t invent it. By stretching commonsense to breaking point someone might possibly argue that discovering the PROOF that proves the validity of a theorem is actually an “invention,” but if you said that Andrew Wiles “invented” his proof of Fermat’s Last Theorem you would be laughed at for years and rightly so. The comparison given with a map is is simplistic to the point of craziness - produce an equation that describes, say, a street scene, make up your own variables.Belfast_{August 24, 2020
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BobRyan: There would be no order, since order cannot arise out of chaos. Who said the universe started out as chaos? I don't think anyone says such a thing.JVL_{August 24, 2020
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In general I agree with Seversky; the question about whether mathematics is discovered or invented is almost purely philosophical and you can easily 'do' or apply mathematics without worrying about its origins at all. I believe the notation, the order we discover theorems, the problems that guide what work is done are historical and cultural. But I believe the underlying mathematics is constant, always. I think it has to be. But, that is just my opinion. I think the map analogy is pretty good: a map contains arbitrary and abstract symbols which are placeholders for other quantities and objects. Not all of them physical by the way: a one-way traffic indicator is separate from the line indicating a street or road and represents an aspect of the road that is not inherent in the road itself. Variables in mathematical equations represent unknowns which are usually fairly well constrained to a particular value or function or operator; they you try and figure out which values or functions or whatever satisfy the equation. So, the analogy is not perfect but a map and an equation are both arbitrary, abstract representations of other things.JVL_{August 24, 2020
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@ Seversky @ 2 You haven’t a clue what you are talking about, Seversky. Mathematics was not “invented” – it was discovered and aspects are still being discovered. And it is nothing at all like a map - a mathematical model contains relationships (that what the equals, =, sign shows) and it contains variables., neither of which are present in the common understanding of a map.Belfast_{August 24, 2020
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Seversky @ 2 The universe works a certain way, since it was designed to work in a certain way. Without design, nothing would work. There would be no order, since order cannot arise out of chaos.BobRyan_{August 23, 2020
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Gödel's incompleteness theorems should be considered as well. The result is that we choose the math that seems to be the most applicable. Then, we amaze ourselves at how precisely it models the physics of a clockwork universe. And finally, we find unsettling and irreconcilable anomalies that force us to choose different math so we first enter a science fantasy phase. Rinse and repeat. Just saying. -QQuerius_{August 23, 2020
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Think about it a different way. The Universe works in certain ways and we have invented a very flexible and powerful language which we can use to model how it works. Like a map, it abstracts key elements of what we observe and uses them to create a model of what is there and just as the map is not the territory so is a model not the same as what is being modeled.Seversky_{August 23, 2020
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I encourage everyone interested in this subject to read this great (and short) paper on it: On the Logic of Being and Wigner's Astonishment Regarding the Applicability of Mathematicsjohnnyb_{August 23, 2020
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