Responding to Ed George About Mathematics

_{William Murray

December 13, 2018

Intelligent Design}

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In another thread, Ed George insists that humans invented mathematics as a way to describe the behavior of phenomena, but that doesn’t mean mathematics is an intrinsic aspect of the universe, a part we discovered, not invented. Here’s why that position is untenable.

Mr. George is correct that humans invent languages – the language of mathematics included. Languages are systems of symbols that represent things. For example, the word “sphere” can be expressed with different symbols in different languages, but the symbols all refer to the same thing – in this case, the form of an object in the real world. That we invented the symbols and language to describe a real thing doesn’t mean we invented the real thing itself.

As Mr. George agrees, mathematics (in terms of this debate) is an invented system of symbols used to describe behaviors of phenomena (physics).

However, humans did not invent those behaviors; we are only describing them using symbolic language. Phenomena in the universe behave in, let’s say, “X” manner. X is a set of discoverable patterns. We discovered those patterns and applied symbolic language to represent and calculate them. In the same way that “sphere-ness” is an inherent quality of something in the universe which we use the term “sphere” to represent, “mathematics” is a term we use to represent an inherent quality of the universe.

Yet, Mr. George denies that we can know whether or not we “discovered” these behaviors (which we call “mathematics”. Of course we did, and we use symbolic language to describe those qualities and behaviors we have discovered.

This same, simple logic can be applied more broadly. We invented a symbolic language in order to refer to things we discover about our existence and the universe, as KF is pointing out, in terms of logical first principles. We did not invent that 1+2=3; those symbols represent observable facts. We did not invent the principle of identity out of whole cloth; it represents an observable fact and, more deeply, a universal structure that human minds cannot escape, no matter how hard we try or imagine. As KF points out, it is responsible for our ability to have cognition at all or to invent and use language. Logical first principles are a fact of our existence which we discovered – first as “X”, then using a string of symbols to represent.

Beyond observable facts, such symbolic language can represent other discoverable facts; such as, some things are impossible to imagine. Imagine that 1+2=4 in any observable way. You can say the words or write the equation, but it is not possible to imagine it being a discoverable fact in any scenario. It’s a nonsensical proposition, much like a 4-sided triangle. The inability to imagine a thing has other implications, but that’s for another conversation.

Language is the invention, but language is itself governed by certain necessary rules. Those rules were entirely hidden to us in the beginning, but we know they were there because inevitably all languages follow those fundamental rules even if we are unaware of them, the first of which is the principle of identity. Without that, language is impossible.

These “X” characteristics of our universe and our existence are things we discovered and then used symbolic systems to represent.

Comments

Hazel and GE, we are not principally discussing particular laws or theories of science (though the exactitude of quantum results gives pause) but the quantities and structure of the world that renders it and its constituents substantially mathematical. That said, WJM is right to point out that mathematical description and frameworks have an astonishingly good fit that is also suggestive. We already saw that for any particular possible world W, it must manifest distinct identity so that W = {A|~A} which you cannot effectively deny on pain of finding indistinguishable things to be identical. Once that is present, nullity, unity and duality are, thence by successive order types the natural numbers thus also transfiniteness. From such the additive inverses, rationals and power series of ratios required for irrationals yield the continuum, which is manifest in space. Thence, displacement, its rate of change and its second rate give us trajectories of accelerated motion. All of this is before we can try to express a law. We see structure and quantity inextricably and pervasively embedded in the world. That is, the substance of mathematics. All of this has been pointed out, you have had little or no answer apart from verbal shifts and declaration of differences of opinion. That pattern speaks, and not in favour of the objections made. KFkairosfocus_{December 14, 2018
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Hazel@39. I like it. I think that his quote is an accurate reflection of our use of mathematics to model reality. It may work very well in many cases, but it is not perfect. As we would expect for any human endeavour.Ed George_{December 14, 2018
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A quote of Einstein's to ponder: "As far as the laws of mathematics refer to reality, they are not certain; and as far as they are certain, they do not refer to reality."hazel_{December 14, 2018
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Honestly, if anyone thought that finding a perfect sphere or cube would be a sign of intelligent design, how can they not see the perfectly mathematical behavior of phenomena as such? It's so bizarre that we think of objects that don't follow precise mathematical behaviors as being something in further need of explanation. It's like requiring further explanation for any rough spheroid shape of rock we find because it's NOT a perfect sphere.William J Murray_{December 14, 2018
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hazel, So then you can say mathematical properties exist in the real world, just as you can say "treeness" properties exist in the real world. It really seems, from this perspective, that some people are just trying to generate a semantic or conceptual dodge. The mathematical precision of behaviors in the real world is a real thing - as real and precise as if we found a perfect sphere in the real world. Continue the line of reasoning and terms you have started. It would be one thing if phenomena behaved in a "sort-of" mathematical way; but they do not. They behave with incredible mathematical precision. So, if you're going to commit to "circularity" or "treeness" being an actual quality (represented via symbolic language) that is actual and inherent in things in the universe, then even more so should you find "mathematics" an actual, inherent quality of the real world, because we're not just finding "sphereness" in these equations, so to speak - we've found perfect spheres (precise, perfectly mathematical behaviors). It's not just "order" we are looking at; it's wondrous order of incredible, mathematical precision. There's literally no other way to describe it. When you look at these behaviours, you might as well be looking at the physical manifestation of the mathematical equation. The equation was there for us to discover, we did not invent it. What we did was create a symbolic language with which to represent that equation. We did not invent the equation that acceleration is inversely proportional to mass given the same force; that mathematical equation existed, whether we expressed that equation in one language or another. We just discovered it. If mathematics wasn't sewn into the fabric of the universe as a discoverable feature (whether on purpose or not), why would that equation **always** hold true? Or any physics equation? We're not finding "mathematics-ness" like "sphere-ness"; the universe expresses perfect, precise mathematics.William J Murray_{December 14, 2018
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An analogy is a TV cooking show. We can say the cook is cracking an egg, mixing a cake, baking in the oven, etc. However, if we were to dissect the signals emanating from the screen, none of those elements described would be found in the signals. Do we then conclude the cook, egg, cake and oven do not exist?EricMH_{December 14, 2018
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Hazel:
But in what sense is “treeness” real? It’s not real in the same sense that a particulur tree is real. It’s real as an abstraction about the physical world, but it doesn’t seem to me that it exists in the physical world . . . . As I think about this, I understand, I think, that no one is claiming that abstract concepts themselves are part of the physical world. But other than the fact that the world is such that mathematical concepts can accurately describe and model or map the world, I’m not sure what else can be said. If the concepts aren’t in the physical world, what is there that could be called “math” that is in the physical world?
Here we come up against our struggle to move beyond a hidden premise of the materialistic age: real implies physical. The evidence is, that things can be that are not physical, and the evidence is that intelligence-bearing information, mind and mathematical frameworks affect and constrain or are even expressed in physical reality but are not themselves equal to signals or bodies. One hint is that quantum weirdness seems to be suspiciously similar. KFkairosfocus_{December 14, 2018
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As I think about this, I understand, I think, that no one is claiming that abstract concepts themselves are part of the physical world. But other than the fact that the world is such that mathematical concepts can accurately describe and model or map the world, I'm not sure what else can be said. If the concepts aren't in the physical world, what is there that could be called "math" that is in the physical world? I think, as Rob Sheldon alluded to above, this is a perennially slippery subject in which, in a Gestaltish way, one can slide back and forth between a Platonic and an Aristotelian view. Or even from a Platonic view, between seeing the Platonic ideals as mirror-like descriptions of the world as opposed to somehow causally informing the word. Interesting topic, but probably mostly a philosophical point-of-view issue without any way to settle the issue.hazel_{December 14, 2018
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Thanks. I understand that "treeness" is the set of properties that real trees have, and that there is a group of real objects, with real properties, that are properly classified as a "tree". But in what sense is "treeness" real? It's not real in the same sense that a particulur tree is real. It's real as an abstraction about the physical world, but it doesn't seem to me that it exists in the physical world. So when you write that
Does “circularity” exist? Yes, if by that we define it as “the properties that all circles have”
I agree. I just can't grasp that that is the same as saying circularity exists in the real world. The real physical world exists with a practically infinite amount of detail: it just is what it is. It is describable by math, and this is a wonderful fact about it, but that is different than saying the abstract concepts we use to describe it are actually part of the physical world. The individual properties characteristics of the world are physically real, but abstractions such as "the properties that all circles have (or trees)" are not physically real. That's the way it seems to me, I guess.hazel_{December 14, 2018
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Ed George gives primacy to what is 'real' to the physical universe, and holds that the mathematics discovered by the mind of man that, (thus far as far as our best scientific testing of the mathematical predictions will allow), perfectly describe the universe, are just an imaginary invention of man's mind, not a discovery. Yet, common sense (and quantum mechanics) gives primacy to what is 'real' to the mind of man, and even to the Mind of God, that 'invented' mathematics, not to the universe.
“The principal argument against materialism is not that illustrated in the last two sections: that it is incompatible with quantum theory. The principal argument is that thought processes and consciousness are the primary concepts, that our knowledge of the external world is the content of our consciousness and that the consciousness, therefore, cannot be denied. On the contrary, logically, the external world could be denied—though it is not very practical to do so. In the words of Niels Bohr, “The word consciousness, applied to ourselves as well as to others, is indispensable when dealing with the human situation.” In view of all this, one may well wonder how materialism, the doctrine that “life could be explained by sophisticated combinations of physical and chemical laws,” could so long be accepted by the majority of scientists." – Eugene Wigner, Remarks on the Mind-Body Question, pp 167-177. He goes toe-to-toe with science big wigs… and so far he’s undefeated. - interview Dr. Bernardo Kastrup: You see we always start from the fact that we are conscious. Consciousness is the only carrier of reality and existence that we can know. Everything else is abstraction; [they] are inferences we make from consciousness. http://www.skeptiko.com/274-bernardo-kastrup-why-our-culture-is-materialistic/ "In any philosophy of reality that is not ultimately self-defeating or internally contradictory, mind – unlabeled as anything else, matter or spiritual – must be primary. What is “matter” and what is “conceptual” and what is “spiritual” can only be organized from mind. Mind controls what is perceived, how it is perceived, and how those percepts are labeled and organized. Mind must be postulated as the unobserved observer, the uncaused cause simply to avoid a self-negating, self-conflicting worldview. It is the necessary postulate of all necessary postulates, because nothing else can come first. To say anything else comes first requires mind to consider and argue that case and then believe it to be true, demonstrating that without mind, you could not believe that mind is not primary in the first place." - William J. Murray “No, I regard consciousness as fundamental. I regard matter as derivative from consciousness. We cannot get behind consciousness. Everything that we talk about, everything that we regard as existing, postulates consciousness.” Max Planck (1858–1947), the main founder of quantum theory, The Observer, London, January 25, 1931 “Consciousness cannot be accounted for in physical terms. For consciousness is absolutely fundamental. It cannot be accounted for in terms of anything else.” Schroedinger, Erwin. 1984. “General Scientific and Popular Papers,” in Collected Papers, Vol. 4. Vienna: Austrian Academy of Sciences. Friedr. Vieweg & Sohn, Braunschweig/Wiesbaden. p. 334.
Advancements in Quantum Mechanics, (a thoroughly mathematical theory with blatantly spiritual overtones), have now experimentally verified this 'common sense' conclusion that mind must be primary for anything else to be real. Specifically, advances in quantum mechanics have now falsified 'realism', which is the belief that the universe exists independently from mind, apart from any conscious observation.
Should Quantum Anomalies Make Us Rethink Reality? Inexplicable lab results may be telling us we’re on the cusp of a new scientific paradigm By Bernardo Kastrup on April 19, 2018 Excerpt: ,, according to the current paradigm, the properties of an object should exist and have definite values even when the object is not being observed: the moon should exist and have whatever weight, shape, size and color it has even when nobody is looking at it. Moreover, a mere act of observation should not change the values of these properties. Operationally, all this is captured in the notion of “non-contextuality”: ,,, since Alain Aspect’s seminal experiments in 1981–82, these predictions (of Quantum Mechanics) have been repeatedly confirmed, with potential experimental loopholes closed one by one. 1998 was a particularly fruitful year, with two remarkable experiments performed in Switzerland and Austria. In 2011 and 2015, new experiments again challenged non-contextuality. Commenting on this, physicist Anton Zeilinger has been quoted as saying that “there is no sense in assuming that what we do not measure [that is, observe] about a system has [an independent] reality.” Finally, Dutch researchers successfully performed a test closing all remaining potential loopholes, which was considered by Nature the “toughest test yet.”,,, It turns out, however, that some predictions of QM are incompatible with non-contextuality even for a large and important class of non-local theories. Experimental results reported in 2007 and 2010 have confirmed these predictions. To reconcile these results with the current paradigm would require a profoundly counterintuitive redefinition of what we call “objectivity.” And since contemporary culture has come to associate objectivity with reality itself, the science press felt compelled to report on this by pronouncing, “Quantum physics says goodbye to reality.” The tension between the anomalies and the current paradigm can only be tolerated by ignoring the anomalies. This has been possible so far because the anomalies are only observed in laboratories. Yet we know that they are there, for their existence has been confirmed beyond reasonable doubt. Therefore, when we believe that we see objects and events outside and independent of mind, we are wrong in at least some essential sense. A new paradigm is needed to accommodate and make sense of the anomalies; one wherein mind itself is understood to be the essence—cognitively but also physically—of what we perceive when we look at the world around ourselves. https://blogs.scientificamerican.com/observations/should-quantum-anomalies-make-us-rethink-reality/ An experimental test of non-local realism - 2007 Simon Gröblacher, Tomasz Paterek, Rainer Kaltenbaek, Caslav Brukner, Marek Zukowski, Markus Aspelmeyer & Anton Zeilinger Abstract: Most working scientists hold fast to the concept of ‘realism’—a viewpoint according to which an external reality exists independent of observation. But quantum physics has shattered some of our cornerstone beliefs. According to Bell’s theorem, any theory that is based on the joint assumption of realism and locality (meaning that local events cannot be affected by actions in space-like separated regions) is at variance with certain quantum predictions. Experiments with entangled pairs of particles have amply confirmed these quantum predictions, thus rendering local realistic theories untenable. Maintaining realism as a fundamental concept would therefore necessitate the introduction of ‘spooky’ actions that defy locality. Here we show by both theory and experiment that a broad and rather reasonable class of such non-local realistic theories is incompatible with experimentally observable quantum correlations. In the experiment, we measure previously untested correlations between two entangled photons, and show that these correlations violate an inequality proposed by Leggett for non-local realistic theories. Our result suggests that giving up the concept of locality is not sufficient to be consistent with quantum experiments, unless certain intuitive features of realism are abandoned. http://www.nature.com/nature/journal/v446/n7138/full/nature05677.html Do we create the world just by looking at it? - 2008 Excerpt: In mid-2007 Fedrizzi found that the new realism model was violated by 80 orders of magnitude; the group was even more assured that quantum mechanics was correct. Leggett agrees with Zeilinger that realism is wrong in quantum mechanics, but when I asked him whether he now believes in the theory, he answered only “no” before demurring, “I’m in a small minority with that point of view and I wouldn’t stake my life on it.” For Leggett there are still enough loopholes to disbelieve. I asked him what could finally change his mind about quantum mechanics. Without hesitation, he said sending humans into space as detectors to test the theory.,,, (to which Anton Zeilinger responded) When I mentioned this to Prof. Zeilinger he said, “That will happen someday. There is no doubt in my mind. It is just a question of technology.” Alessandro Fedrizzi had already shown me a prototype of a realism experiment he is hoping to send up in a satellite. It’s a heavy, metallic slab the size of a dinner plate. http://seedmagazine.com/content/article/the_reality_tests/P3/ Experimental non-classicality of an indivisible quantum system - Zeilinger 2011 Excerpt: Page 491: "This represents a violation of (Leggett's) inequality (3) by more than 120 standard deviations, demonstrating that no joint probability distribution is capable of describing our results." The violation also excludes any non-contextual hidden-variable model. The result does, however, agree well with quantum mechanical predictions, as we will show now.,,, https://vcq.quantum.at/fileadmin/Publications/Experimental%20non-classicality%20of%20an%20indivisible.pdf Albert Einstein vs. Quantum Mechanics and His Own Mind – video https://www.youtube.com/watch?v=vxFFtZ301j4
in the following experiment, that was performed with atoms instead of photons, it was proved that measurement is everything. At the quantum level, reality does not exist if you are not looking at it,"
Experiment confirms quantum theory weirdness - May 27, 2015 Excerpt: The bizarre nature of reality as laid out by quantum theory has survived another test, with scientists performing a famous experiment and proving that reality does not exist until it is measured. Physicists at The Australian National University (ANU) have conducted John Wheeler's delayed-choice thought experiment, which involves a moving object that is given the choice to act like a particle or a wave. Wheeler's experiment then asks - at which point does the object decide? Common sense says the object is either wave-like or particle-like, independent of how we measure it. But quantum physics predicts that whether you observe wave like behavior (interference) or particle behavior (no interference) depends only on how it is actually measured at the end of its journey. This is exactly what the ANU team found. "It proves that measurement is everything. At the quantum level, reality does not exist if you are not looking at it," said Associate Professor Andrew Truscott from the ANU Research School of Physics and Engineering. Despite the apparent weirdness, the results confirm the validity of quantum theory, which,, has enabled the development of many technologies such as LEDs, lasers and computer chips. The ANU team not only succeeded in building the experiment, which seemed nearly impossible when it was proposed in 1978, but reversed Wheeler's original concept of light beams being bounced by mirrors, and instead used atoms scattered by laser light. "Quantum physics' predictions about interference seem odd enough when applied to light, which seems more like a wave, but to have done the experiment with atoms, which are complicated things that have mass and interact with electric fields and so on, adds to the weirdness," said Roman Khakimov, PhD student at the Research School of Physics and Engineering. http://phys.org/news/2015-05-quantum-theory-weirdness.html
The Theistic implications of this experiment are fairly obvious. As Professor Scott Aaronson quipped, “Look, we all have fun ridiculing the creationists,,, But if we accept the usual picture of quantum mechanics, then in a certain sense the situation is far worse: the world (as you experience it) might as well not have existed 10^-43 seconds ago!”
“Look, we all have fun ridiculing the creationists who think the world sprang into existence on October 23, 4004 BC at 9AM (presumably Babylonian time), with the fossils already in the ground, light from distant stars heading toward us, etc. But if we accept the usual picture of quantum mechanics, then in a certain sense the situation is far worse: the world (as you experience it) might as well not have existed 10^-43 seconds ago!” – Scott Aaronson – MIT associate Professor quantum computation - Lecture 11: Decoherence and Hidden Variables
Of related note:
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.,,, Universes do not “spontaneously create” on the basis of abstract mathematical descriptions, nor does the fantasy of a limitless multiverse trump the explanatory power of transcendent intelligent design. What Mr. Hawking’s contrary assertions show is that mathematical savants can sometimes be metaphysical simpletons. Caveat emptor. http://www.washingtontimes.com/news/2010/oct/1/hawking-irrational-arguments/
Verse:
Colossians 1:17 He is before all things, and in him all things hold together.
bornagain77_{December 14, 2018
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Hazel, not only does treeness exist but key properties involved are informational, encoded in a language, in dna. Codes, as we know, are massively structured and quantitative phenomena, so are mathematical.KFkairosfocus_{December 14, 2018
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hazel: Trees are real in the world, but is “treeness”? Yes. "Treeness" is the real set of properties that real trees have. What you're really talking about is classification. Another way to refer to "treeness" is that there is a class of real objects we call "tree" that have a certain set of real properties. But "treeness" is certainly a valid shorthand (and folksy) way of describing this set of properties that trees have. Does "circularity" exist? Yes, if by that we define it as "the properties that all circles have."mike1962_{December 14, 2018
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I agree in part with what wjm says, but there is one distinction I am unclear about. Mathematics uses symbolic representations to describe real things. The particular systems of symbolic representations we use are human inventions. For instance, the famous difference between the notations for derivatives offered by Newton and Leibnitz, where we finally settled on Leibnitz's. Also, the world obviously has structure, order, and regularities which can be described mathematically. I really like the quote in the OP: ”Mystery number one is how is it that the physical world does in fact accord with mathematics, and not just any mathematics but very sophisticated, subtle mathematics to such a fantastic degree of precision." Here is the part I am unclear on. wjm says,
The symbols contained in mathematics are symbolic representations of those real things – like the word “water” is symbolic of the real thing, or the term “tree” is a symbolic representation of the real thing. It makes no more sense to say that humans invented mathematics than it makes sense to say that humans invented water or trees.
There are real specific trees. However the word "tree" is an abstract concept that does not represent any specific real tree, but applies to a set of general characteristics which are present in specific trees. So in whatever sense the concept represented by the word "tree" is real, it is real in a different way than actual trees are real. We didn't invent trees. We did invent the word "tree". Trees are real in the world, but is "treeness"? Applying these thoughts to math (and I'm sure the situation is different, but I'm not sure how), spherical things exists, and the concept of sphere can represent a certain common property (all points equidistant from a center point). However, real spheres are never perfectly spherical, and they all have properties other than their sphericalness. So in what sense is the mathematical concept of sphere actually in the real world?hazel_{December 14, 2018
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Did Fibonacci invent the Fibonacci sequence? The Fibonacci sequence is “a series of numbers in which each number ( Fibonacci number ) is the sum of the two preceding natural numbers”: 0+1=1;1+1=2; 1+2=3; 2+3=5; 3+5=8; 5+8=13; 8+13=21… (forever) 1, 2,3,5,8,13,21… are the Fibonacci numbers. Some really interesting things happen when you apply this sequence to geometry by tiling the numbers: https://en.wikipedia.org/wiki/Fibonacci_number#/media/File:FibonacciSpiral.svg This seems to be nothing more than a clever parlor trick, so it seems like something that Fibonacci invented.* Right? But if that’s true why do we find the Fibonacci sequence in nature? Take a look at the first ten minutes of this episode of Nova, which is a regular science program on PBS. The episode gives several examples of where the Fibonacci sequence as well as the value of pi appear, sometime quite unexpectedly, in nature. It’s followed by an interview with MIT physicist Max Tegmark who believes that everything in nature can be reduced to mathematics. He appears to suggest that we could all be living in some sort of virtual reality. https://www.youtube.com/watch?v=mpcpzXuzdQk The whole documentary is worth watching because it is examining the very same question that we are considering here. *Actually Indian mathematicians "discovered" the sequence about 1200 years earlier.john_a_designer_{December 14, 2018
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kf@25, Well said. What I always find most interesting is the cognitive dissonance on display. Here we have someone who agrees that what mathematics is describing are real, natural aspects of the world, but will not agree that mathematics is a real, natural aspect of the world. Yes, we all know the word itself, the symbolic representation itself, the model itself is not the reality, but that's true of every word or phrase or model we use to represent the reality. To say "humans invented mathematics" is therefore the precise equivalent to saying "humans invented inertia" or "humans invented horses." I mean, there are a lot of things that humans invented, but the mathematical behavior of aspects of the universe, and as a foundational aspect of identity and cognitive thought, is not among those inventions. Some of those discoverable aspects of existence and the universe are so ubiquitous, like identity and gravity, it takes a very keen mind to even notice them and a very stubborn mind to ignore or deny once revealed.William J Murray_{December 14, 2018
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hazel said:
There is lots of math that doesn’t model anything real. Does that make it useless? Or have I missed something in the preceding conversation about this?
Any language can be used to describe both real and unreal things. That's a completely irrelevant point. Nobody is trying to claim that the symbolic representations themselves are real in the sense that they are discoverable - we didn't go out and find "2+3=5" in a natural rock formation, just as we didn't find the English world "rock" or "tree" in the rock or in the tree. Does inertia exist in the real world? Does gravity? Does electromagnetic radiation? Does the way phenomena behave in relationship to those things actually exist in the real world? We discovered those things and the related behaviors. We applied symbolic language to describe them. It makes no more sense to say "we invented mathematics" than it makes sense to say "we invented inertia, gravity and electromagnetic radiation." It's like saying we invented the fact that if you add two apples to three apples you have five apples. No, we did not - we started using a specific language to represent an experiential fact. As with all language terms that represent real things, mathematics represents a real thing we find in the universe. ALL such words and terms are representational.William J Murray_{December 14, 2018
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WJM, I am inclined to hold that in our studies we do bring to bear cultural features, but the focal substance is structure and quantity. So, reality has a mathematical aspect and we may investigate it as a project of our civilisation. The discipline is not the substance, and the discipline recognises the substance. Indeed, even as "horse" recognises certain noble animals, "one," or 1 or I etc recognise the reality of unity. Similarly "two" etc recognise that there is duality, inevitably in any distinct possible world. Our challenge, I believe, is that the Kantian ugly gulch has now pervaded our culture and we think we are locked up in a world of appearances, shared memes etc that cannot bridge to things in themselves. That is, we have discredited truth and are beginning to fall into chaos. The loss of ability to recognise the manifest reality of say maleness and femaleness leading to 112 "genders" at last count, is but one symptom of that chaos. Far more central is what we are seeing, radical relativisation and cultural captivity of science and mathematics that locks them away from seeking and being motivated and governed by truth, accurate description of reality. The question haunts me: has our culture gone suicidally insane, caught up in Plato's Cave shadow-shows? KFkairosfocus_{December 14, 2018
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ES @ 10: "objective and complete knowledge of the world"? That seems to be perilously close to omniscience! Did you ever find any scientists who were that inclined to play at being God? I would think that seeking observationally reliable, substantial and useful understanding and knowledge of the world tempered by due awareness of our epistemic limitations would be enough for creatures such as we are. At the same time, I find it highly significant that knowledge requires acknowledgement of what is well-warranted, i.e. belief, that seemingly bitter pill to swallow. As in, well-warranted (so, reliable), credibly true belief. Without truth-seeking and humility before truth, however, the moral mainspring of knowledge is broken. So, I am disinclined to reduce science and mathematics to modelling, which explicitly surrenders truth-seeking in interests of utility and simplicity. We may use models in such disciplines and we must be aware that error exists but truth -- accurate description of reality -- must ever be a goal and chief virtue. KFkairosfocus_{December 14, 2018
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Ed George said:
I don’t get your point. Why would we attempt to model something that isn’t real?
Mr. George, do you not understand that the term "mathematics", and the formulas that describe the behavior of phenomena, are symbolic representations of something you agree is REAL? You admit that what mathematics represents is real. The symbols contained in mathematics are symbolic representations of those real things - like the word "water" is symbolic of the real thing, or the term "tree" is a symbolic representation of the real thing. It makes no more sense to say that humans invented mathematics than it makes sense to say that humans invented water or trees.William J Murray_{December 14, 2018
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EG @ 4: The issue isn't conclusion jumping but conclusion resisting! There is no question but that the world has in it features or aspects that are quantitative or structural, which is the substance of mathematics. Further, distinct identity of any possible world directly entails duality, unity, nullity thence the natural numbers, i.e. countable quantity. So, it's not a happenstance of our world, it is framework for any world. The existence of a world also inevitably requires order instead of utter chaos, i.e. structure. Again, the substance of mathematics. Also, that distinct stable identity of a world implies that events, circumstances and objects within it will reflect core ordering characteristics so we can ground inductive exploration and discovery of such rational principles, aka laws; despite our error-prone epistemic limitations, we are not reduced to despair that we can never cross the ugly gulch to understand (in part at least) things as they are. That is empirical, reliable, well-warranted and even accurate -- truthful -- knowledge of the observed cosmos is possible. Further, we have in fact found a framework of such laws, which are intensely mathematical, and exhibit fine tuning that sets our world at a deeply isolated operating point in the space of mathematically possible frameworks. And yes, that strongly points to design of the cosmos by an intelligent agent whose mind is also inclined to mathematics; though that is not our primary focus here. KFkairosfocus_{December 14, 2018
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ES, yes our observations are error prone but they also often hit the mark: e.g. there are exactly four letters in m-1 a-2 r-3 k-4; this is of course a quantitative phenomenon. Likewise, rational reflection and insights reveal self-evident first truths and principles. Identity being central, and opening up how quantity and structure (the substance as opposed to the study of Mathematics) are inescapably integral to the framework for any possible world. KFkairosfocus_{December 13, 2018
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H, the issue is not all of Mathematics or of its extensions into abstract logic model worlds, but a key core. That embedding starts with distinct identity and its correlate, distinction thus duality, unity, nullity. From this the von Neumann succession of order types directly brings out the naturals. Which, lead to the integers, the rationals, the reals and many properties and manifestations. For example falling implies continually changed location thus length and the continuum. It also brings in infinitesimals and/or limits (thus, power series) as we address rates dx/dt --> v, dv/dt --> a, F = dP/dt, thus F = ma when m is constant. We cannot even properly observe and measure the phenomena without recognising the centrality of structure and quantity. Which, is the substance (as opposed to the study) of Mathematics. Going further, there is a tendency to trivialise the pervasiveness of pi and e in all sorts of phenomena, and the way e, pi, i, 0 and 1 are dovetailed to infinite precision in the Euler expression. Properties such as primeness and irrationality (e.g. sqrt-2, which popped up in pondering the ratio of sides to diagonal of a square) come out. While we are on the Pythagoreans, they noticed how we instinctively respond to whole number ratios in music, indeed the ratios were found at the root of pleasing musical patterns -- transforming how we understand aesthetics. There is much more, all pointing to how quantity and structure are embedded in our cosmos and indeed any possible world. Which on massive evidence invites study and is integral to the intelligibility of the world. Which in turn is why we find quantitative laws, theories and models so useful. All of which then highlights the need for a more insightful and more accurate understanding. KFkairosfocus_{December 13, 2018
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There is lots of math that doesn't model anything real. Does that make it useless? Or have I missed something in the preceding conversation about this? I see, I think, after looking back a few points. The issue is whether the math is actually real in the world, as opposed to in our symbolic system. The argument seems to be that if the model fits the real world so well, as it does, then it makes sense to say the math is in the world and not just i the model. Do I understand your point, Mike?hazel_{December 13, 2018
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Mike1962
What other way is there to look at it? The mathematics are useless unless the thing it actually models is real. (And does so with astonishing precision.)
I don’t get your point. Why would we attempt to model something that isn’t real?Ed George_{December 13, 2018
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Euler's identity is neat, and is one of many marvelous math facts, but I don't think Euler's identity is embedded in the universe anyplace, nor models any specific phenomena. I might be wrong, though, and would welcome being corrected by an example.hazel_{December 13, 2018
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Eugene S: Any such description is bound to be limited in accuracy and to include some cognitive bias. As soon as there is an epistemic cut between the observer and the observed, some subjectivity is there. Therefore scientific knowledge has the disturbing property of never being able to deliver complete knowledge. I can agree with that, however, e, Pi, and i appear over and over again in our best and extremely accurate models and physical descriptions, and are curiously related as described by Euler's Identity. How is that true if those values, and that relation, are not baked into the universe in a fundamental way?mike1962_{December 13, 2018
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Ed George: ...to jump to the conclusion that, because mathematics is good at modelling the universe, that mathematics is inherent in the universe is not warranted. What other way is there to look at it? The mathematics are useless unless the thing it actually models is real. (And does so with astonishing precision.) Pi shows up everywhere. E shows up everywhere. i shows up everywhere. Quantum physics, the most precise set of descriptions and means of prediction the world has seen, depends on the use of all of these. As does General Relativity and a hosts of other physical phenomena. And before quantum physics came along, Euler proved the cozy relation of e^ i pi + 1 = 0. Whoa Nellie! So if these mathematical facts are not baked into the universe, why does the universe act like they are? With astonishing precision. It is possible that our model is not completely accurate, but so far there is no indication of it. And even if were are somewhat wrong, something very close to what the mathematics describes would have to be true... long before humans invented mathematics.mike1962_{December 13, 2018
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Moreover, the anomalies that are now found in the “Cosmic Microwave Background Radiation” (CMBR) (which the perfect flatness of the universe allows us to study in detail), highlights just how unique our solar system and earth are in this universe,,
In other words, the “tiny temperature variations” in the CMBR, (from the large scale structures in the universe, to the earth and solar system themselves), reveal teleology, (i.e. a goal directed purpose, a plan), that specifically included the earth from the start. ,,, The earth, from what our best science can now tell us, is not a random cosmic fluke as atheists presuppose. https://uncommondescent.com/intelligent-design/our-solar-system-is-a-lot-rarer-than-it-was-a-quarter-century-ago/#comment-669546
Besides the flatness of the universe, "Platonic perfection" is also found in other places in physics. One little known 'platonic perfection' is the higher dimensional "amplituhedron" of Quantum-electrodynamics that was found a few years back:
Bohemian Gravity - Rob Sheldon - September 19, 2013 Excerpt: Quanta magazine carried an article about a hypergeometric object that is as much better than Feynman diagrams as Feynman was better than Heisenberg's S-matrices. But the discoverers are candid about it, "The amplituhedron, or a similar geometric object, could help by removing two deeply rooted principles of physics: locality and unitarity. “Both are hard-wired in the usual way we think about things,” said Nima Arkani-Hamed, a professor of physics at the Institute for Advanced Study in Princeton, N.J., and the lead author of the new work, which he is presenting in talks and in a forthcoming paper. “Both are suspect.”" What are these suspect principles? None other than two of the founding principles of materialism--that there do not exist "spooky-action-at-a-distance" forces, and that material causes are the only ones in the universe.,,, http://rbsp.info/PROCRUSTES/bohemian-gravity/
And the other place(s) where 'platonic perfection' is found is in the 4-Dimensional space time curvature of special and general relativity. Simply put, no experimental test to date has ever been able to detect any 'imperfection' for what the theory(s) of relativity predict (for higher dimensional 4-D space-time curvature and/or geometry). As Berlinski noted,
“On the other hand, I disagree that Darwin’s theory is as `solid as any explanation in science.; Disagree? I regard the claim as preposterous. Quantum electrodynamics is accurate to thirteen or so decimal places; so, too, general relativity. A leaf trembling in the wrong way would suffice to shatter either theory. What can Darwinian theory offer in comparison?” (Berlinski, D., “A Scientific Scandal?: David Berlinski & Critics,” Commentary, July 8, 2003)
Supplemental note:
Quantum Mechanics, Special Relativity, General Relativity and Christianity - video https://www.youtube.com/watch?v=h4QDy1Soolo
bornagain77_{December 13, 2018
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EugeneS mentions “imperfections” in spheres, (and supposedly also imperfections in triangles, squares, lines, etc.. etc..), and indeed there are no perfect spheres (nor any other perfect Euclidean objects in this universe). It is also interesting to note the history that assuming perfection has had in science. Copernicus, (who was heavily influenced by Platonic thinking), imagined (incorrectly) that the planets move in perfect circles (rather than ellipses). Later, Newton, for allowing God could adjust the orbits of the planets, was chastised by Leibniz, (and Laplace) for having a "very narrow ideas about the wisdom and the power of God.".. i.e. For having a narrow view of the perfection of God. Laplace, contrary to atheistic folklore, cited with approval Leibniz's criticism of Newton's invocation of divine intervention to restore order to the Solar System: "This is to have very narrow ideas about the wisdom and the power of God.", to them, it would count as evidence against intelligent design if God had to intervene to prevent the solar system from collapsing. So intelligent design could just as easily be a motivation to prove the stability of the solar system.
"Leibniz, in his controversy with Newton on the discovery of infinitesimal calculus, sharply criticized the theory of Divine intervention as a corrective of the disturbances of the solar system. "To suppose anything of the kind", he said, "is to exhibit very narrow ideas of the wisdom and power of God'." - Pierre-Simon Laplace https://books.google.com/books?id=oLtHAAAAIAAJ&pg=PA73&lpg=PA73
Moreover, contrary to what is commonly believed, Laplace did not really solve the problem of planetary perturbations in the end, (he only solved for for first degree approximations), but Haret showed that orbits are not absolutely stable using third degree approximations. Moreover, I hold that if Newton and Leibniz (and even Laplace) could see our science today they would be very pleased by what modern science has now revealed about the wisdom and power of God:
“You might also think that these disparate bodies are scattered across the solar system without rhyme or reason. But move any piece of the solar system today, or try to add anything more, and the whole construction would be thrown fatally out of kilter. So how exactly did this delicate architecture come to be?” R. Webb - Unknown solar system 1: How was the solar system built? - New Scientist – 2009 Is the Solar System Stable? By Scott Tremaine - 2011 Excerpt: So what are the results? Most of the calculations agree that eight billion years from now, just before the Sun swallows the inner planets and incinerates the outer ones, all of the planets will still be in orbits very similar to their present ones. In this limited sense, the solar system is stable. However, a closer look at the orbit histories reveals that the story is more nuanced. After a few tens of millions of years, calculations using slightly different parameters (e.g., different planetary masses or initial positions within the small ranges allowed by current observations) or different numerical algorithms begin to diverge at an alarming rate. More precisely, the growth of small differences changes from linear to exponential:,,, As an example, shifting your pencil from one side of your desk to the other today could change the gravitational forces on Jupiter enough to shift its position from one side of the Sun to the other a billion years from now. The unpredictability of the solar system over very long times is of course ironic since this was the prototypical system that inspired Laplacian determinism. Fortunately, most of this unpredictability is in the orbital phases of the planets, not the shapes and sizes of their orbits, so the chaotic nature of the solar system does not normally lead to collisions between planets. However, the presence of chaos implies that we can only study the long-term fate of the solar system in a statistical sense, by launching in our computers an armada of solar systems with slightly different parameters at the present time—typically, each planet is shifted by a random amount of about a millimeter—and following their evolution. When this is done, it turns out that in about 1 percent of these systems, Mercury’s orbit becomes sufficiently eccentric so that it collides with Venus before the death of the Sun. Thus, the answer to the question of the stability of the solar system—more precisely, will all the planets survive until the death of the Sun—is neither “yes” nor “no” but “yes, with 99 percent probability.” https://www.ias.edu/about/publications/ias-letter/articles/2011-summer/solar-system-tremaine
Along this line of "perfection" thought, it is interesting to note where in this universe perfection for spheres is approached rather closely,,,
Sun's Almost Perfectly Round Shape Baffles Scientists - (Aug. 16, 2012) — Excerpt: The sun is nearly the roundest object ever measured. If scaled to the size of a beach ball, it would be so round that the difference between the widest and narrow diameters would be much less than the width of a human hair.,,, They also found that the solar flattening is remarkably constant over time and too small to agree with that predicted from its surface rotation. http://www.sciencedaily.com/releases/2012/08/120816150801.htm Bucky Balls - Andy Gion Excerpt: Buckyballs (C60; Carbon 60) are the roundest and most symmetrical large molecule known to man. Buckministerfullerine continues to astonish with one amazing property after another. C60 is the third major form of pure carbon; graphite and diamond are the other two. Buckyballs were discovered in 1985,,, http://www.3rd1000.com/bucky/bucky.htm
The delicate balance at which carbon is synthesized in stars is truly a work of art.,,, Years after Sir Fred discovered the stunning precision with which carbon is synthesized in stars he stated this:
"I do not believe that any physicist who examined the evidence could fail to draw the inference that the laws of nuclear physics have been deliberately designed with regard to the consequences they produce within stars." Sir Fred Hoyle - "The Universe: Past and Present Reflections." Engineering and Science, November, 1981. pp. 8–12
And perfection for a sphere is also approached in the Cosmic Background Radiation (CBR). ,,, Of the supposed "imperfections" in the sphere of the CBR, the following author comments, "the discovery of small deviations from smoothness (anisotopies) in the cosmic microwave background is welcome, for it provides at least the possibility for the seeds around which structure formed in the later Universe"
The Cosmic Background Radiation Excerpt: These fluctuations are extremely small, representing deviations from the average of only about 1/100,000 of the average temperature of the observed background radiation. The highly isotropic nature of the cosmic background radiation indicates that the early stages of the Universe were almost completely uniform. This raises two problems for (a naturalistic understanding of) the big bang theory. First, when we look at the microwave background coming from widely separated parts of the sky it can be shown that these regions are too separated to have been able to communicate with each other even with signals traveling at light velocity. Thus, how did they know to have almost exactly the same temperature? This general problem is called the horizon problem. Second, the present Universe is homogenous and isotropic, but only on very large scales. For scales the size of superclusters and smaller the luminous matter in the universe is quite lumpy, as illustrated in the following figure. ,,, Thus, the discovery of small deviations from smoothness (anisotopies) in the cosmic microwave background is welcome, for it provides at least the possibility for the seeds around which structure formed in the later Universe. However, as we shall see, we are still far from a quantitative understanding of how this came to be. http://csep10.phys.utk.edu/astr162/lect/cosmology/cbr.html
It is also interesting to note where "platonic perfection" is, not only approached, but arguably reached in the universe.,,, The universe is perfectly flat as far as our best scientific instruments can tell us:
How do we know the universe is flat? Discovering the topology of the universe - by Fraser Cain - June 7, 2017 Excerpt: With the most sensitive space-based telescopes they have available, astronomers are able to detect tiny variations in the temperature of this background radiation. And here's the part that blows my mind every time I think about it. These tiny temperature variations correspond to the largest scale structures of the observable universe. A region that was a fraction of a degree warmer become a vast galaxy cluster, hundreds of millions of light-years across. The cosmic microwave background radiation just gives and gives, and when it comes to figuring out the topology of the universe, it has the answer we need. If the universe was curved in any way, these temperature variations would appear distorted compared to the actual size that we see these structures today. But they're not. To best of its ability, ESA's Planck space telescope, can't detect any distortion at all. The universe is flat.,,, We say that the universe is flat, and this means that parallel lines will always remain parallel. 90-degree turns behave as true 90-degree turns, and everything makes sense.,,, Since the universe is flat now, it must have been flat in the past, when the universe was an incredibly dense singularity. And for it to maintain this level of flatness over 13.8 billion years of expansion, in kind of amazing. In fact, astronomers estimate that the universe must have been flat to 1 part within 1×10^57 parts. Which seems like an insane coincidence. https://phys.org/news/2017-06-universe-flat-topology.html
Interestingly, this 'perfect flatness' is essential for us to be able to practice math and science,
Why We Need Cosmic Inflation By Paul Sutter, Astrophysicist | October 22, 2018 Excerpt: As best as we can measure, the geometry of our universe appears to be perfectly, totally, ever-so-boringly flat. On large, cosmic scales, parallel lines stay parallel forever, interior angles of triangles add up to 180 degrees, and so on. All the rules of Euclidean geometry that you learned in high school apply. But there’s no reason for our universe to be flat. At large scales it could’ve had any old curvature it wanted. Our cosmos could’ve been shaped like a giant, multidimensional beach ball, or a horse-riding saddle. But, no, it picked flat. https://www.space.com/42202-why-we-need-cosmic-inflation.html
bornagain77_{December 13, 2018
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Robert Sheldon,
It’s an old debate. “Do we discover math, or invent it?” Plato says we discover it. Aristotle says we invent it. Most mathematicians who have advanced beyond algebra, would be Platonists. Even those who claim to be Aristotelians will admit they are functionally Platonists.
According to mathematician Roger Penrose, who collaborated with Stephen Hawking on some of his early work, “mathematics seems to have its own kind of existence.” He then goes on to explain:
It is very important in understanding the physical world that our way of describing the physical world, certainly at its most precise, has to do with mathematics. There is no getting away from it. That mathematics has to have been there since the beginning of time. It has eternal existence. Timelessness really. It doesn’t have any location in space. It doesn’t have any location in time. Some people would take it not having a location with not having any existence at all. But it is hard to talk about science really without giving mathematics some kind of reality because that is how you describe your theories in terms of mathematical structures… It also has this relationship to mentality because we certainly have access to mathematical truths. I think it is useful to think of the world as not being a creation of our minds because if we do then how could it have been there before we were around? If the world is obeying mathematical laws with extraordinary precision since the beginning of time, well, there were no human beings and no conscious beings of any kind around then. So how can mathematics have been the creation of minds and still been there controlling the universe? I think it is very valuable to think of this Platonic mathematical world as having its own existence. So let’s allow that and say that there are three different kinds of existence. There may be others, but three kinds of existence: the normal, physical existence; the mental existence (which seems to have, in some sense, an even greater reality – it is what we are directly aware of or directly perceive); and the mathematical world which seems to be out there in some sense conjuring itself into existence – it has to be there in some sense.
https://www.reasonablefaith.org/media/reasonable-faith-podcast/roger-penrose-interview-part-1/#_ftn3 Just to clarify, earlier in the interview Penrose described his metaphysical world view as a tripartite one consisting of the physical world, the mental world and a separate and distinct mathematical world. He goes on to explain that… ’there is the relationship between these three worlds which I regard, all three of them, as somewhat mysterious or very mysterious. I sometimes refer to this as “three worlds and three mysteries.” Mystery number one is how is it that the physical world does in fact accord with mathematics, and not just any mathematics but very sophisticated, subtle mathematics to such a fantastic degree of precision. That’s mystery number one.’john_a_designer_{December 13, 2018
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