Either science or naturalism will win

_{Denyse O'Leary

April 27, 2016

Intelligent Design, Mathematics}

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The response to Barry’s recent post, How Did Mathematics Come to be Woven Into the Fabric of Reality? pretty much lays bare the issues: “Why is the universe we live in connected by an unreasonably beautiful, elegant and effective mathematical structure?”

The naturalist says we just evolved to see the world as making sense, but don’t really know. Defending Darwin means that much to him …

At 73, I noted,

Aleta at 81: Sorry, that won’t work. Either man is observing something that underlies the universe and working within those given results or mathematics is an adaptation for survival (it could be completely irrational and is only tangentially related to how the universe works).

The second position saves naturalism (and its creation story, Darwinism) at the price of science. But naturalists are increasingly eager to dump falsifiability and evidence-based thinking. Many are cool with criminalizing dissent.

Science will pick up where it left off, with a hiatus of naturalism somewhere in between.

See also: Tyson bombshell: Universe likely just computer sim

But who needs reality-based thinking anyway? Not the new cosmologists

and

In search of a road to reality

Note: When respectable naturalists sneer at ID, they often use emblems like Blake’s designer god Urizen (above), with the mathematical compasses. Maybe they are trying to tell us more than we realize. They sneer at an emblem of mathematics because it grounds perceptions in evidence. If they are theistic naturalists, their God is so great that he doesn’t need evidence or even to make sense, or in the end, even to exist. We wouldn’t really know if he did or not. That works well wth a lot of new religious movements.

Comments

Byers: Our math is not gods math OK but here we are talking about mathematics, what humans understand it to be. If you understand something about God's math that some of us don't, that's fine, I'm just not privy to all aspects of a mind that would be God's, and would never seek to be. I do however find the idea that God would need or want to measure something, or anything, or all things to be kind of pointless to discuss on a thread about science and mathematics.groovamos_{April 29, 2016
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groovamos, you write,
but I have actually seen online where mathematicians have actually rued the term “imaginary” as a descriptor, because the complex domain is so foundational to math. When I was in high school I thought the idea of something imaginary in math to be really obscure, but it is not.
This comment bears on the central issue about the nature of math that has been floating around for a few threads. I used to give a little enrichment mini-lecture on this to my pre-calculus class. It went something like this: What we call the real numbers are called "real" because they have been slowly developed by creating different types of numbers which model various aspects of the real world. Abbreviating greatly, we started with counting numbers, obviously. Fractions came next - here, have half a cow. Interesting enough, irrationals came next, as the Pythagorean Theorem produced the sqrt(2), and it was proved that could not be any rational number. And over 1000 years later zero was introduced, first as a placeholder in the decimal number system. Negative numbers were next in the western world. The Chinese, Hindus, and Arabs had worked with them, and accepted them as meaningful to various degrees, but western mathematicians resisted them as meaningless - not real, up until the 1600's. See https://en.wikipedia.org/wiki/Negative_number#History for a quick overview of their history. However, negative numbers were finally accepted for three reasons: 1. People worked out the rules for arithmetic with them, and found that they were internally consistent. For instance, if you consider the quadratic equation x^2 + 2x - 3 = 0, get the answer x = –3, and then put –3 back in for x in the original equation, it works. This is algebra I stuff now, but the greatest mathematicians in the western world were uncertain about accepting this 500 years ago. 2. People found real-world applications for negative number: gains and losses, debts owed and paid, etc. 3. People (Descartes, for instance) visualized the negative numbers on a number line stretching backwards from 0. So at this point, all the numbers on the number line were considered "real" because we knew how they worked, could apply them to the real world, and could visualize them as a coherent, continuous set on a line extending indefinitely in both directions. But we still had unsolvable equations! x^2 = –4 has no real solution, since no number, positive or negative, times itself is negative. So somebody decided, boldly, to make up a new kind of number, sqrt(-1), and because it could not be a "real" number, they called it imaginary, and used the symbol i. This was a bad move. It turns out that you can repeat the three steps above: 1. Rules for doing arithmetic with i are easy. 2. It was found that imaginary numbers could be visualized on a number line perpendicular to the real number line, with a common origin. (I won't tell the story of why that is so here, but obviously I did with my classes.) 3. Eventually many real-world uses were found for i, and for the extension to complex numbers which combine real and imaginary numbers. So, and this is the important conclusion, i is no more, and no less, imaginary or real than 1 is. 1 is more basic, and easier to see in the real world, than i is. But i is not more real in respect to the world of pure math than i is. All numbers have the same status as a concept in the coherent, integrated world of pure math. Numbers are abstract concepts which we map onto aspects of the real world at times in order to describe the world. But some numbers aren't more real just because they yield more easily to being used as a description of some aspect of reality. groovamos writes,
Congrats BTW on using polar coordinates and trig to prove what I posted. How did you know to do that?
Thanks. I usd to teach the basics of this to my calculus class. The last couple of days of class, after the last test, I culminated the year explaining why e^(i•pi) + 1 = 0 is true. I used the infinite series for sine, cosine, and e to show that e^(ix) = cos x + i sin x, and showed how that could be interpreted on the complex plane. From there, Euler's Identity is obvious. I pointed out that Euler's Identity is just a number fact: they started their public school math education with 1 + 1 = 2 in kindergarten and ended it with e^(i•pi) + 1 = 0 as seniors, but both are just true facts about numbers. And P.S., we also talked about the Mandelbrot set, and I showed them a little about how it is created using iterative functions on the complex plane. So that is how I knew to show that your problem did equal zero.Aleta_{April 28, 2016
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grooveamos the bible says this and it makes sense to me. Measurement is not a special case. If the universe is measured by God and so constructed. then all this math suff is just a special case in the grand measurement. Why not? so our math is our attempts to understand this grand measure. yet our math is just a language or special case within the true measure. If he universe was measured out then it would be grand. Our math is not gods math.Robert Byers_{April 28, 2016
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Aleta : Does this make sense to you, and possibly clear up a confusion? Or do you still see a problem with what I said? Well not exactly a problem, but I have actually seen online where mathematicians have actually rued the term "imaginary" as a descriptor, because the complex domain is so foundational to math. When I was in high school I thought the idea of something imaginary in math to be really obscure, but it is not. Nature requires it. For example why would nature care if you were to say that your wall outlet provides -60 Hz power. Nature requires that it make as much sense as 60 Hz because it takes two counter rotating vectors in the complex plane to define a sinusoidal time function. This is no joke, or trick question, it is related in the same to the integral transforms which map the time domain to complex domains. Congrats BTW on using polar coordinates and trig to prove what I posted. How did you know to do that? Speaking of trig, I'm going to credit my post for including '3' for the following reasons: '3' does not show up in analysis nearly as much as say sqrt(2) or even '2' and you hardly ever see it. But look at how subtle it is: Consider that 3 is the root of the words trigonometry, triangle, triangular, triangulation. 3 is the closest integer to pi and e and lies between them. Our 3-phase power was based on the necessity for power generating equipment to have theoretically constant power load possible. Single phase alternators at industrial scale would vibrate at 120 Hz and literally autoshake to destruction. Nikola Tesla was the first to visualize the solution as ideal smooth loading based on the following. Square 3 sinusoids identical in amplitude and frequency spaced 120 degrees apart, and sum: sin^2(at) + sin^2(at + 2*pi/3) + sin^2(at - 2*pi/3) = 3 'at' can be seen as a time function or set to zero for proof of the above. Also sin^2(x) is sin(x)^2 The 3 on the RHS is viewed as power, and is constant, so the alternator will not autoshake. Never seen the word 'autoshake' BTW but '3' is very subtle in math, not showing up explicitly in analysis, yet being so foundational to trig.groovamos_{April 28, 2016
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Yes. For one thing, people can be taught. I could teach someone with a high school math background why that identity is true, and what it all means.Aleta_{April 28, 2016
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Aleta,
Possibly just for fun, possibly to see if I understood complex numbers.
In any case, I was a bit taken aback by the discouraging tone of the last sentence in groovamos' post. :-)daveS_{April 28, 2016
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Possibly just for fun, possibly to see if I understood complex numbers.Aleta_{April 28, 2016
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Edit (previous version is now moot): I don't know if this post of groovamos' is intended to be a response to something Aleta said. The connection is unclear.daveS_{April 28, 2016
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e^(-3i*pi/4) + e^(3i*pi/4) + 2/2^(1/2) - edited to add parenthesesZachriel_{April 28, 2016
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I can't quite get that to work out: [cos (-135) + sin (-135)] + [cos (135) + sin (135)] + 1/sqrt(2) = 2 cos (135) + sqrt(2)/2 = - sqrt(2) + sqrt(2)/2 = - sqrt(2)/2??? What am I missing? [Edit: I see that you changed it from 1/(2^1/2) to 2/(2^1/2) while I was working on it, and now it works out.]Aleta_{April 28, 2016
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An equation incorporating 1, 0, i, e, pi, the all-important 2 AND its square AND its square root plus a bonus of the frequent 3, the sum 1+2: e^-3i*pi/4 + e^3i*pi/4 + 2/(2^1/2) = 0 Have not seen this stated anywhere before I visualized in bed this morning but if you are far into adulthood and don't see why it's true then you will never understand complex variables.groovamos_{April 28, 2016
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to groovamos: I think you are misunderstanding my use of the word "applied", perhaps. The things you are saying about the uses of complex numbers are also what I was referring to: we use complex numbers to describe the basic nature of reality, and they are of fundamental importance. Here is the distinction that I think is confusing. Often we contrast theoretical vs applied science. Theoretical physics, for instance, is interesting in understanding how things work irrespective of whether there is any practical application. Applied science then uses what is known for practical purposes. Often theoretical research later yields practical uses, but that is not its main purpose. That is not the meaning of applied I was using. I was, instead, contrasting pure mathematics, which is developed solely within the context of mathematics itself, with applying math to understanding the real world via making and testing mathematical models. When imaginary numbers were first invented, around 1600 or so, most mathematicians considered them nonsense (hence imaginary), but even after they were accepted mathematically no one had any idea how they could be applied to understanding the world. However, as time went by, they were shown to be tremendously useful in describing the world, as you mentioned. So my use of the word "applied" was in this second sense, contrasting math used to describe the word with math that exists purely in the realm of math. I was not contrasting theoretical vs applied in the first sense, as it is often used in science. Does this make sense to you, and possibly clear up a confusion? Or do you still see a problem with what I said?Aleta_{April 28, 2016
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Byers: The bible is clear that God has measured out the universe. It says it repeatedly. My complaint witrh mATH is that its not the true measuement or we don’t know it. instead its a language of the measurement. Its not invented by man however. its discovered. however the discoveries are not the actual conclusion of the measurement. Possibly;y special cases of it. Boy that's some seriously dogma-fueled dreaming there. Go start reading up on estimation theory in wikipedia if you want to bring measurement into the conversation otherwise you're just broadcasting your ignorance of the topic. Mathematics is about objects and quantities, sets and groups and operators. Measurement is studied in a narrow special branch of statistics called estimation theory. There is a branch of mathematics (of which I'm totally ignorant) called measure theory that has absolutely nothing to do with your bathroom scales or your tape measure.groovamos_{April 27, 2016
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Aleta : Then later, once the math of complex numbers was more completely developed, they were found to be tremendously useful in applied contexts Highly misleading and almost wrong. Complex numbers are fundamental to analysis and I have pointed out using the most obvious explanations before on this board and I'm going to do it again. Without complex variables, the trig functions are just 4, 6, and 7 letter words to help with trig static measurements and problems and tables. But even the tables would not exist if the above were true. The trig functions are fundamental to reality as the exponentially damped sinusoid (sine or cosine functions of time and space) of a photon propagating through vacuum and any non-ferromagnetic material shows. And in the analysis of the above, sinusoidal functions appear as eigenfunctions in the solutions of linear differential equations describing the behavior of the phenomena. And the only way sinusoidal functions (and all trig functions) show up in the descriptions of reality (analysis) are as pairs of complex conjugate time function powers of e. That is it. No other way. They can only be understood in the context of the Taylor's series expansions of the trig functions, which is an outgrowth of the calculus. This is not merely an understanding for "applied context". The complex numbers are fundamental to the behavior of nature, and to any description of such. Especially in physics and electrical engineering, also in the study of any dynamic system involving energy oscillatory or dissipative modes, say in mechanical engineering but also in non-energy modal systems such as in discrete-time signal processing by computation (digital signal processors). Also complex variables are fundamental to all of the integral transforms. The Laplace transform, the Fourier and Z transforms cannot be grasped at an advanced level without a firm understanding of complex variables at an intermediate level at least.groovamos_{April 27, 2016
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The bible is clear that God has measured out the universe. It says it repeatedly. My complaint witrh mATH is that its not the true measuement or we don't know it. instead its a language of the measurement. Its not invented by man however. its discovered. however the discoveries are not the actual conclusion of the measurement. Possibly;y special cases of it. the order in the universe is proof of a creator. the math is a poor representation of this order. or we do a poor job of it. however man did not create math. And its only been 6000 years since we were counting things up.Robert Byers_{April 27, 2016
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Hi News. I explained why I thought your response to my thoughts on math were not particularly on target. I also explained a bit to ba77 at 7 that I felt that the discussion about math was metaphysically neutral, so I have no idea why you wrote, "Defending Darwin means that much to him …", since there is nothing I've written that has anything to do with Darwin. Would you care to comment on any of the points I've made about math?Aleta_{April 27, 2016
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Despite the attempted downplaying by atheists, both Wigner and Einstein are on record as to regarding it as a epistemological 'miracle' that humans can reliably model and describe the world mathematically:
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 "You find it strange that I consider the comprehensibility of the world (to the extent that we are authorized to speak of such a comprehensibility) as a miracle or as an eternal mystery. Well, a priori, one should expect a chaotic world, which cannot be grasped by the mind in any way .. the kind of order created by Newton's theory of gravitation, for example, is wholly different. Even if a man proposes the axioms of the theory, the success of such a project presupposes a high degree of ordering of the objective world, and this could not be expected a priori. That is the 'miracle' which is constantly reinforced as our knowledge expands." Albert Einstein - Letters to Solovine - New York, Philosophical Library, 1987
Einstein's quote "a priori, one should expect a chaotic world, which cannot be grasped by the mind in any way" is particularly telling. In other words, Atheists have no idea why the fine-tuned universal constants remain constant and why we can reliably model the world mathematically. In fact, since atheists ultimately presuppose a random/chaotic basis as the creator of reality instead of God as the creator and sustainer of reality, atheists are continually 'surprised' whenever we find that the constants have not varied over the entire history of the universe. For instance:
Dark energy alternatives to Einstein are running out of room – January 9, 2013 Excerpt: Last month, a group of European astronomers, using a massive radio telescope in Germany, made the most accurate measurement of the proton-to-electron mass ratio ever accomplished and found that there has been no change in the ratio to one part in 10 million at a time when the universe was about half its current age, around 7 billion years ago. When Thompson put this new measurement into his calculations, he found that it excluded almost all of the dark energy models using the commonly expected values or parameters. If the parameter space or range of values is equated to a football field, then almost the whole field is out of bounds except for a single 2-inch by 2-inch patch at one corner of the field. In fact, most of the allowed values are not even on the field. “In effect, the dark energy theories have been playing on the wrong field,” Thompson said. “The 2-inch square does contain the area that corresponds to no change in the fundamental constants, (a 'true cosmological constant'), and that is exactly where Einstein stands.” http://phys.org/news/2013-01-dark-energy-alternatives-einstein-room.html Hugh Ross PhD. - Scientific Evidence For Cosmological Constant (1 in 10^120 Expansion Of The Universe) video 23:12 minute mark https://youtu.be/fTP01yi-SSU?t=1392
Here is the paper from the atheistic astrophysicists, that Dr. Ross referenced in the preceding video, that speaks of the ‘disturbing implications’ of the finely tuned expanding universe (1 in 10^120 cosmological constant):
Disturbing Implications of a Cosmological Constant - Dyson, Kleban, Susskind (each are self proclaimed atheists) - 2002 Excerpt: "Arranging the universe as we think it is arranged would have required a miracle.,,," “The question then is whether the origin of the universe can be a naturally occurring fluctuation, or must it be due to an external agent which starts the system out in a specific low entropy state?” page 19: “A unknown agent [external to time and space] intervened [in cosmic history] for reasons of its own.,,,” Page 21 "The only reasonable conclusion is that we don't live in a universe with a true cosmological constant". http://arxiv.org/pdf/hep-th/0208013.pdf
Here are the verses from the Bible which Dr. Ross listed, which were written well over 2000 years before the discovery of the finely tuned expansion of the universe, that speak of God 'Stretching out the Heavens'; Job 9:8; Isaiah 40:22; Isaiah 44:24; Isaiah 48:13; Zechariah 12:1; Psalm 104:2; Isaiah 42:5; Isaiah 45:12; Isaiah 51:13; Jeremiah 51:15; Jeremiah 10:12. The following verse is my favorite out of the group of verses:
Job 9:8 He alone stretches out the heavens and treads on the waves of the sea. The Truman Show – Truman walking on water – screenshot picture http://gaowsh.files.wordpress.com/2011/03/screen-shot-2011-03-29-at-5-09-50-pm-2.jpg
Moreover, it is also important to note just how small of a change in the universal constants, as atheists presuppose, it would take to destroy our ability to reliably mathematically model the universe and to therefore destroy our ability to practice science
Scientists Question Nature’s Fundamental Laws – Michael Schirber – 2006 Excerpt: “There is absolutely no reason these constants should be constant,” says astronomer Michael Murphy of the University of Cambridge. “These are famous numbers in physics, but we have no real reason for why they are what they are.” The observed differences are small-roughly a few parts in a million-but the implications are huge (if they hold up): The laws of physics would have to be rewritten, not to mention we might need to make room for six more spatial dimensions than the three that we are used to.”,,, The speed of light, for instance, might be measured one day with a ruler and a clock. If the next day the same measurement gave a different answer, no one could tell if the speed of light changed, the ruler length changed, or the clock ticking changed. http://www.space.com/2613-scientists-question-nature-fundamental-laws.html
Although the atheist has no reason why he believes the constants of the universe should remain constant, the Theist rightly expects the constants of the universe to remain unchanging.
Psalm 119:89-91 Your eternal word, O Lord, stands firm in heaven. Your faithfulness extends to every generation, as enduring as the earth you created. Your regulations remain true to this day, for everything serves your plans.
At the 28:09 minute mark of the following video, Dr Hugh Ross speaks of the 7 places in the bible that speak of unchanging universal constants.
Symposium 2015 : Scientific Evidence For God's Existence - Hugh Ross - video https://youtu.be/4mEKZRm1xXg?t=1689
In fact, modern science was born out of the Christian worldview by men of devout Christian faith precisely because of their expectation of unchanging universal constants:
“Men became scientific because they expected Law in Nature, and they expected Law in Nature because they believed in a Legislator. In most modern scientists this belief has died: it will be interesting to see how long their confidence in uniformity survives it.” Lewis, C.S., Miracles: a preliminary study, Collins, London, p. 110, 1947.
Personally, I find it to be a source of great comfort that the universal constants do not vary. The reason is this: since God's faithfulness is reflected in the unchanging nature of the universal constants then I can also trust that God will not break his promises to us, ever! Verse and Music
John 14:2-3 "In My Father's house are many dwelling places; if it were not so, I would have told you; for I go to prepare a place for you. "If I go and prepare a place for you, I will come again and receive you to Myself, that where I am, there you may be also." Hillsong UNITED - Touch The Sky - Official Lyric Video - HD https://www.youtube.com/watch?v=T-mALvKKCe0
bornagain77_{April 27, 2016
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Aleta, Yes, I had forgotten that Conway's game had come up in this thread. That would definitely count as an invention. Coincidentally, when typing my post about convex functions above, I had first considered using another Conway invention, the thrackle, instead.daveS_{April 27, 2016
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Dave, are you familiar with Conway's game of Life? He invented this, and experimented with different sets of rules before settling on the current one. This led to a whole world of interesting mathematics, with results that can only be derived by going step-by-step through the iterations. It's fascinating. See Wikipedia and other googled places if interested. The Game of Life has to count as an invention, I think.Aleta_{April 27, 2016
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“In so far as a scientific statement speaks about reality, it must be falsifiable; and in so far as it is not falsifiable, it does not speak about reality.” Karl Popper – The Two Fundamental Problems of the Theory of Knowledge (2014 edition), Routledge Darwinian Evolution is a Unfalsifiable Pseudo-Science – Mathematics – video https://www.facebook.com/philip.cunningham.73/videos/vb.100000088262100/1132659110080354/?type=2&theater It’s (Much) Easier to Falsify Intelligent Design than Darwinian Evolution – Michael Behe, PhD https://www.youtube.com/watch?v=_T1v_VLueGk Conservation of information, evolution, etc - Sept. 30, 2014 Excerpt: Kurt Gödel’s logical objection to Darwinian evolution: "The formation in geological time of the human body by the laws of physics (or any other laws of similar nature), starting from a random distribution of elementary particles and the field is as unlikely as the separation of the atmosphere into its components. The complexity of the living things has to be present within the material [from which they are derived] or in the laws [governing their formation]." Gödel - As quoted in H. Wang. “On `computabilism’ and physicalism: Some Problems.” in Nature’s Imagination, J. Cornwall, Ed, pp.161-189, Oxford University Press (1995). Gödel’s argument is that if evolution is unfolding from an initial state by mathematical laws of physics, it cannot generate any information not inherent from the start – and in his view, neither the primaeval environment nor the laws are information-rich enough.,,, More recently this led him (Dembski) to postulate a Law of Conservation of Information, or actually to consolidate the idea, first put forward by Nobel-prizewinner Peter Medawar in the 1980s. Medawar had shown, as others before him, that in mathematical and computational operations, no new information can be created, but new findings are always implicit in the original starting points – laws and axioms.,,, http://potiphar.jongarvey.co.uk/2014/09/30/conservation-of-information-evolution-etc/ Evolutionary Computing: The Invisible Hand of Intelligence - June 17, 2015 Excerpt: William Dembski and Robert Marks have shown that no evolutionary algorithm is superior to blind search -- unless information is added from an intelligent cause, which means it is not, in the Darwinian sense, an evolutionary algorithm after all. This mathematically proven law, based on the accepted No Free Lunch Theorems, seems to be lost on the champions of evolutionary computing. Researchers keep confusing an evolutionary algorithm (a form of artificial selection) with "natural evolution." ,,, Marks and Dembski account for the invisible hand required in evolutionary computing. The Lab's website states, "The principal theme of the lab's research is teasing apart the respective roles of internally generated and externally applied information in the performance of evolutionary systems." So yes, systems can evolve, but when they appear to solve a problem (such as generating complex specified information or reaching a sufficiently narrow predefined target), intelligence can be shown to be active. Any internally generated information is conserved or degraded by the law of Conservation of Information.,,, What Marks and Dembski (mathematically) prove is as scientifically valid and relevant as Gödel's Incompleteness Theorem in mathematics. You can't prove a system of mathematics from within the system, and you can't derive an information-rich pattern from within the pattern.,,, http://www.evolutionnews.org/2015/06/evolutionary_co_1096931.html "Either mathematics is too big for the human mind, or the human mind is more than a machine." Kurt Gödel As quoted in Topoi : The Categorial Analysis of Logic (1979) by Robert Goldblatt, p. 13 "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 Russell Wallace, New Thoughts on Evolution, 1910 the intellect (is) immaterial and immortal. If today’s naturalists do not wish to agree with that, there is a challenge for them. ‘Don’t tell me, show me’: build an artificial intelligence system that imitates genuine mathematical insight. There seem to be no promising plans on the drawing board.,,, James Franklin is professor of mathematics at the University of New South Wales in Sydney.bornagain77_{April 27, 2016
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PS to my #6: I alluded to definitions in my post #3, but upon further reflection, I think I probably downplayed their significance as an invented component of mathematics. I don't know much about the Mandelbrot set, but I suspect its "discovery" spurred the formation of new definitions, thereby influencing future discoveries and inventions. I remember chancing on a book many years ago on convex functions and wondering how someone could actually get an entire book out of that. I was surprised to see that it's actually a vast (and very useful) field of research. Apparently whoever defined the concept of "convex function" made a very good decision, one which certainly wasn't logically necessary, so I think that should count as an invention.daveS_{April 27, 2016
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to ba77: I think all the points I'm making about math are not related to whether God exists or not, or whether materialism is true. We may disagree about how the universe became as it is, what the nature and source of man's rationality may be, and whether math has some Platonic existence in the mind of God, but I think pretty much everything I've written in 2 and 5 above could be accepted by a theist mathematician.Aleta_{April 27, 2016
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Aleta,
In theory, all of math may be discovered, but in practice it has taken a great deal of human invention to discover it.
Yes, good point. I think no one would say that a novel (written in English, say, with vocabulary all contained in the OED) was "discovered" rather than invented or created, even though my exact argument about the power set of C could be adapted to cover it as well. Obviously, there is a vast difference in complexity (of the Kolmogorov variety, I suppose) between a novel and the Mandelbrot set, but functionally, there is a great deal of ingenuity/creativity/"invention" in the practice of mathematics.daveS_{April 27, 2016
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Thanks, Dave. And yes, perhaps the dichotomy between invention and discovery can be nuanced a bit. The beautiful thing about math is that the logic compels further results. Once we set the basic foundations, to use your term, the results can unravel before our eyes, with a life of their own, so to speak. In that sense, we discover the embedded logical consequences. This is true of pure math, irrespective of any application or relationship to the physical world. On the other hand, we often make further "inventions" along the way, so the interplay between invention and discovery is continual I think. Example: both Leibnitz and Newton invented some basic concepts and definitions that are the foundation of calculus. Of course calculus, in theory, is a "discovery" that flows from the very foundations of the set theory of numbers, but it certainly is not an obvious, direct consequence that didn't require some considerable invention of preceding mathematics to happen. To be more concrete, calculus would probably not have been invented without the prior invention of coordinate geometry by Descartes. That incredibly powerful tool allowed us to visualize functional relationships between two variables, and changed geometry from being the static study of shapes into a dynamic study of motion. Also, both Leibnitz and Newton invented language and notations to discuss the infinitesimal changes that are at the heart of calculus. It turns out that Leibnitz's system was much more useful from a practical point of view than Newton's, and so it's Leibnitz's that we use today. This certainly falls under the category of invention, I think. Also, the actual process of developing math is not a direct path: when we are done and summarize our results, the flow from beginning foundations to end results can look all tidy and inevitable, but in actual practice it's the activity of people going down all sorts of dead ends, discussing and arguing with others, making mistakes, etc. that finally leads to that nice final result. So, to summarize, even though in theory everything in math is a logical consequence of the foundations and logical tools that start the system, in practice a great deal of human inventions is necessary to develop the further mathematical concepts and tools that we need to fully access those logical results. The Mandelbrot set may be, in theory, completely entailed logically by the foundations of number theory, but without people having decided to study iterative functions in the complex plane, having picked the particular function to iterate, and having invented the technology to deal with both the calculations and the visualization, I'm not sure it makes sense to say the Mandlebrot set existed before all the above happened. It existed potentially, but did it "really exist" until we invented the various ideas necessary to formalize it it symbols and pictures? Here's a relevant favorite saying by Yogi Berra: In theory, there is no difference between theory and practice, but in practice there is. In theory, all of math may be discovered, but in practice it has taken a great deal of human invention to discover it.Aleta_{April 27, 2016
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in his Defense of the Divine Revelation against the Objections of the Freethinkers (atheists) the brilliant mathematician Leonhard Euler observed that there are people who are simply incapable of being reached by reason: A DEFENSE OF THE (Divine) REVELATION AGAINST THE OBJECTIONS OF FREETHINKERS, BY MR. EULER Excerpt: "The freethinkers (atheists) have yet to produce any objections that have not long been refuted most thoroughly. But since they are not motivated by the love of truth, and since they have an entirely different point of view, we should not be surprised that the best refutations count for nothing and that the weakest and most ridiculous reasoning, which has so often been shown to be baseless, is continuously repeated. If these people maintained the slightest rigor, the slightest taste for the truth, it would be quite easy to steer them away from their errors; but their tendency towards stubbornness makes this completely impossible." http://www.math.dartmouth.edu/~euler/docs/translations/E092trans.pdf Leonhard Euler (1707-1783) was arguably the greatest mathematician of the eighteenth century God by the Numbers - Connecting the constants Excerpt: The final number comes from theoretical mathematics. It is Euler's (pronounced "Oiler's") number: e^pi*i. This number is equal to -1, so when the formula is written e^pi*i+1 = 0, it connects the five most important constants in mathematics (e, pi, i, 0, and 1) along with three of the most important mathematical operations (addition, multiplication, and exponentiation). These five constants symbolize the four major branches of classical mathematics: arithmetic, represented by 1 and 0; algebra, by i; geometry, by pi; and analysis, by e, the base of the natural log. e^pi*i+1 = 0 has been called "the most famous of all formulas," because, as one textbook says, "It appeals equally to the mystic, the scientist, the philosopher, and the mathematician.",,, The discovery of this number gave mathematicians the same sense of delight and wonder that would come from the discovery that three broken pieces of pottery, each made in different countries, could be fitted together to make a perfect sphere. It seemed to argue that there was a plan where no plan should be.,,, Today, numbers from astronomy, biology, and theoretical mathematics point to a rational mind behind the universe.,,, The apostle John prepared the way for this conclusion when he used the word for logic, reason, and rationality—logos—to describe Christ at the beginning of his Gospel: "In the beginning was the logos, and the logos was with God, and the logos was God." When we think logically, which is the goal of mathematics, we are led to think of God. http://www.christianitytoday.com/ct/2006/march/26.44.html?start=3 Mathematics and Physics – A Happy Coincidence? – William Lane Craig – video https://www.youtube.com/watch?v=BF25AA4dgGg 1. If God did not exist the applicability of mathematics would be a happy coincidence. 2. The applicability of mathematics is not a happy coincidence. 3. Therefore, God exists. William Lane Craig on the unexpected applicability of mathematics to nature - 11/13/13 http://winteryknight.wordpress.com/2013/11/13/william-lane-craig-on-the-unexpected-applicability-of-mathematics-to-nature/bornagain77_{April 27, 2016
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Aleta, Very interesting post. It leads me to consider the notion of "invented mathematics" more carefully. Can we specify precisely the requirements a mathematical invention must satisfy? I don't have a strong argument for this, but it seems to me that perhaps only things such as primitive definitions, axioms, and rules of logic should count as inventions. "Foundations", more briefly. I can only come up with what might appear to be a rather silly illustration: The Mandelbrot set is a subset of the complex plane, so already exists in the power set of C. Once the mathematical substructure leading to the existence of this power set was completed, the Mandelbrot set in some sense was available to be "discovered", rather than invented.daveS_{April 27, 2016
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Well, it's interesting that News chose to highlight this remark. I'd like to unpack her remark, and reply.
Either man is observing something that underlies the universe and working within those given results or mathematics is an adaptation for survival (it could be completely irrational and is only tangentially related to how the universe works).
Yes, our universe is pervaded by mathematical regularities. First, I didn't say, in this remark, or others I've written, that mathematical regularities don't underlie the universe. In fact, the understanding that mathematics is the best tool for describing the physical universe is one of the great cultural realizations that has led to modern science and the modern world. Why is this so? I wrote a long, somewhat rambling post on this question at 59, but the short answer is I don't know why this is so, and I don't think anyone else does either, despite our many varied philosophical ideas about the subject. But I accept the universe as it is, and one of its relevant features is that mathematics can describe it. Next, I think News is quite wrong to state that there is a simple either/or situation: either man observes underlying mathematical regularities or math is an adaption for survival that could be irrational. First, math is a cultural invention. The first written math started no earlier than 10,000 years or so ago, and that was simply invented written symbols for simple counting numbers. I wrote a post about this over on one of the infinity threads, which I encourage interested readers to go look at, because it gives a concrete example about how even the simplest math is man-made. https://uncommondescent.com/atheism/durston-and-craig-on-an-infinite-temporal-past/#comment-600414 So the earliest math, and much math since then, has been stimulated by observing the physical world, but the creation of words and symbols to describe the math is a man-made invention. Math often has grown in response to purely mathematical considerations, not in response to observations of the physical world. I wrote another post on the infinity thread outlining how the history of math can be sketched by seeing it as the progressive invention of new kinds of numbers which extend the number system understood at that time. https://uncommondescent.com/atheism/durston-and-craig-on-an-infinite-temporal-past/#comment-600414 This history highlights an important idea: that even though math often begins by examining the physical world, essentially modeling what we see in the world in abstract, symbolic ways, once math got established, it very often went on the invent new ideas that were not tied to the physical world. For instance, in the post linked to directly above, I briefly point to the history of imaginary numbers (involving sqrt(-1)). These were invented in order to deal with purely mathematical issues that came up in the study of the roots of polynomials: they were not invented in response to any physical phenomena or to solve any applied problem. Then later, once the math of complex numbers was more completely developed, they were found to be tremendously useful in applied contexts Pure and applied math: This example highlights a key issue. Pure mathematics is a formal, abstract system that exists in and of itself, irrespective of any connection to the physical world. It is a man-made cultural product that thousands of people world-wide over especially the last 2500 years have contributed to. One can invent something, like Conway's Game of Life or the mathematics that produce the Mandlebrot set, and then logically develop an incredibly complex set of theorems. Such mathematics often raise interesting ideas that can be related to the phyiscal world, but such mathematics are not about, nor dependent on, the physical world. However, we can also apply math to the physical world. The process of applying math is one of creating a model which maps certain elements of the math to certain elements of the world, and then testing the model by comparing conclusions drawn from the math with actual events in the physical world. If the math and the events agree, that is evidence that the model we propose is an accurate model. If they don't, we modify the model. Note well: the application of math assumes the first point above, which is not in question: that our universe is pervaded by mathematical regularities which our math can describe. But before we can use math to describe the world we have to invent the mathematical tools to do so. Some of those tools are directly motivated by observing the world (for instance, counting), but some are not (for instance, imaginary numbers.) Math as an adaptation for survival: Math is a cultural invention, not an evolutionary adaptation. Mankind's ability to process symbolic understanding via language started tens of thousands of years ago. Irrespective of how it evolved (either with or without intelligent guidance), that ability eventually allowed humans to start on a long road of gaining knowledge about the physical world: language certainly has survival value. But math developed long afterwards: the invention of system for writing language and math were cultural inventions, not evolutionary ones. The evolved ability to process symbols for abstract understanding evolved, but what mankind has built with that ability has been a part of cultural history, not evolutionary history.Aleta_{April 27, 2016
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Thanks for the notice of my remark! :-) I will reply soon ..Aleta_{April 27, 2016
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