Author

Topic: Math: Eternal? If so Who Created It?

Thugpreacha
Member Posts: 12972 From: Denver,Colorado USA Joined: 12302003 Member Rating: 1.2


Message 1 of 30 (851953)
05052019 12:21 PM



I just saw an interesting article from The Huffington Post in 2013. Is Mathematics Invented or Discovered?by Derek Abbott, â€œThe reasonable ineffectiveness of mathematics,â€ Proceedings of the IEEE, Vol. 101, No. 10, pp. 21472153, 2013.quote: Mathematics is the language of science and has enabled mankind to make extraordinary technological advances. There is no question that the logic and order that underpins mathematics, has served us in describing the patterns and structure we find in nature. The successes that have been achieved, from the mathematics of the cosmos down to electronic devices at the microscale, are significant. Einstein remarked, â€œHow can it be that mathematics, being, after all, a product of human thought which is independent of experience, is so admirably appropriate to the objects of reality?â€ Amongst mathematicians and scientists, there is no consensus on this fascinating question. The various types of responses to Einsteinâ€™s conundrum include:1) Math is innate. The reason mathematics is the natural language of science is that the universe is underpinned by the same order. The structures of mathematics are intrinsic to nature. Moreover, if the universe disappeared tomorrow, our eternal mathematical truths would still exist. It is up to us to discover mathematics and its workingsâ€”this will then assist us in building models that will give us predictive power and understanding of the physical phenomena we seek to control. This rather romantic position is what I loosely call mathematical Platonism. 2) Math is a human construct. The only reason mathematics is admirably suited describing the physical world is that we invented it to do just that. It is a product of the human mind and we make mathematics up as we go along to suit our purposes. If the universe disappeared, there would be no mathematics in the same way that there would be no football, tennis, chess or any other set of rules with relational structures that we contrived. Mathematics is not discovered, it is invented. This is the nonPlatonist position. 3) Math is not so successful. Those that marvel at the ubiquity of mathematical applications has perhaps been seduced by an overstatement of their successes. Analytical mathematical equations only approximately describe the real world, and even then only describe a limited subset of all the phenomena around us. We tend to focus on those physical problems for which we find a way to apply mathematics, so overemphasis on these successes is a form of â€œcherry picking.â€ This is the realist position. 4) Keep calm and carry on. What matters is that mathematics produces results. Save the hot air for philosophers. This is called the â€œshut up and calculateâ€ position. The debate over the fundamental nature of mathematics is by no means new and has raged since the time of the Pythagoreans. Can we use our hindsight now to shed any light on the above four positions? A recent development within the last century was the discovery of fractals. Beautiful complex patterns, such as the Mandelbrot set, can be generated from simple iterative equations. Mathematical Platonists eagerly point out that elegant fractal patterns are common in nature, and that mathematicians clearly discover rather than invent them. A counterargument is that any set of rules has emergent properties. For example, the rules of chess are clearly a human contrivance, yet they result in a set of elegant and sometimes surprising characteristics. There are infinite numbers of possible iterative equations one can possibly construct, and if we focus on the small subset that results in beautiful fractal patterns we have merely seduced ourselves. Take the example of infinite monkeys on keyboards. It appears miraculous when an individual monkey types a Shakespeare sonnet. But when we see the whole context, we realize all the monkeys are merely typing gibberish. In a similar way, it is easy to be seduced into thinking that mathematics is miraculously innate if we are overly focused on its successes, without viewing the complete picture. The nonPlatonist view is that, first, all mathematical models are approximations of reality. Second, our models fail, they go through a process of revision, and we invent new mathematics as needed. Analytical mathematical expressions are a product of the human mind, tailored for the mind. Because of our limited brainpower, we seek out compact elegant mathematical descriptions to make predictions. Those predictions are not guaranteed to be correct, and experimental verification is always required. What we have witnessed over the past few decades, as transistor sizes have shrunk, is that nice compact mathematical expressions for ultrasmall transistors are not possible. We could use highly cumbersome equations, but that isnâ€™t the point of mathematics. So we resort to computer simulations using empirical models. And this is how much of cutting edge engineering is done these days. The realist picture is simply an extension of this nonPlatonist position, emphasizing that compact analytical mathematical expressions of the physical world around us are not as successful or ubiquitous as weâ€™d like to believe. The picture that consistently emerges is that all mathematical models of the physical world break down at some point. Moreover, the types of problems addressed by elegant mathematical expressions are a rapidly shrinking subset of all the currently emerging scientific questions. But why does this all matter? The â€œshut up and calculateâ€ position tells us to not worry about such questions. Our calculations come out the same, no matter what we personally believe; so keep calm and carry on. I, for one, believe the question is important. My personal story is that I used to be a Platonist. I thought all mathematical forms were reified and waiting to be discovered. This meant that I philosophically struggled with taking limits to infinity, for example. I merely got used to it and accepted it under sufferance. During my undergraduate days, I had a moment of enlightenment and converted to nonPlatonism. I felt a great burden lift from my shoulders. Whilst this never affected my specific calculations, I believe a nonPlatonist position gives us greater freedom of thought. If we accept that mathematics is invented, rather than discovered, we can be more daring, ask deeper questions, and be motivated to create further change. Remember how irrational numbers petrified the bejesus out of the Pythagoreans? Or the interminable time it took mankind to introduce a zero into arithmetic? Recall the centuries of debate that occurred over whether negative numbers are valid or not? Imagine where science and engineering would be today if this argument was resolved centuries earlier. It is the ravages of Platonistlike thinking that have held back progress. I argue that a nonPlatonist position frees us from an intellectual straightjacket and accelerates progress.~Derek Abbott
It always amuses me when critics claim that even God had to have an origin... yet these same critics won't say the same thing about math! Go figure... Edited by Phat, : No reason given. Chance as a real force is a myth. It has no basis in reality and no place in scientific inquiry. For science and philosophy to continue to advance in knowledge, chance must be demythologized once and for all. ~RC Sproul "A lie can travel half way around the world while the truth is putting on its shoes." ~Mark Twain " ~" If that's not sufficient for you go soak your head."~Faith You can "get answers" by watching the ducks. That doesn't mean the answers are coming from them.~Ringo Subjectivism may very well undermine Christianity. In the same way that "allowing people to choose what they want to be when they grow up" undermines communism.~Stile
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PaulK
Member Posts: 15439 Joined: 01102003 Member Rating: 2.9
(2)




quote:
It always amuses me when critics claim that even God had to have an origin... yet these same critics won't say the same thing about math!
But how often does that happen, really ? You quote quite a bit of discussion, Iâ€™m sure the subject has come up here  but I donâ€™t remember anyone explicitly refusing to say so without good reason. Donâ€™t you really mean that the subject is rarely discussed because it is of less interest ? (Although I will note that since mathematics is abstract while God is supposedly concrete there is a major difference which may be relevant in some philosophical systems.)
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AZPaul3
Member Posts: 4635 From: Phoenix Joined: 11062006 Member Rating: 4.5


Message 3 of 30 (852005)
05052019 11:11 PM



My take. First, math is both discovered and invented. The relationship between a circle and its circumference is a discovery. The invented part is the formula for determining what the circumference of a circle is; the symbolic language humans invented to manipulate and make use of the relationship we discovered. And then we go and confuse the invention of the language with the invention of the relationship. No we did not invent the circle and its circumference. That naturally occurring relationship has always been there well prior to any human having the capacity to even notice there was a relationship. Though the math we use to work the relationship is invented the math that is the relationship is discovered. Second, we should not be surprised at the power of math to model, even if imperfectly, the natural relationships we find in this universe since we inhabit a highly logical, if imperfect, universe. No outside influence was needed to say the circumference of a circle will be this. It is a natural relationship that is an intrinsic part of the structure. It just took a smart ape to discover the math in it and a smarter one to invent the math to manipulate and make use of it. Edited by AZPaul3, : No reason given. Edited by AZPaul3, : No reason given. Eschew obfuscation. Habituate elucidation.
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Sarah Bellum
Member Posts: 413 Joined: 05042019 Member Rating: 1.1

Another interesting aspect of mathematics is the way it demonstrates how an extraordinary degree of complexity arises from extremely simple concepts. Consider, for example, the mathematical concept of a "group", that is, a set with an operation (for example, addition) an identity element (for example, the zero) and a few simple rules (that every element has an inverse, a "negative" element such that if you combine the two you get the identity and other simple principles). Things that can be considered "groups" come in great variety, from the group with two elements {0,1} with the operations 0+0=1 0+1=1 and 1+1=0, to the set of all integers to the set of all permutations of the real numbers. etc. etc. etc. One of the most intriguing is the famous FischerGriess "Monster" group (https://math.berkeley.edu/...whatismonster/whatismonster.pdf http://www.math.s.chibau.ac.jp/~kitazume/monster3.jpg). Such an extraordinary degree of complexity from such simple principles... An important indication of how easy it would be for complex organisms to develop naturally, wouldn't you say?
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RAZD
Member Posts: 20155 From: the other end of the sidewalk Joined: 03142004 Member Rating: 4.0

Welcome to the fray, Sarah Bellum (love the moniker), If you use [url=https://math.berkeley.edu/...whatismonster/whatismonster.pdf http://www.math.s.chibau.ac.jp/~kitazume/monster3.jpg] FischerGriess "Monster" group[/url] you get: FischerGriess "Monster" group or you can try [img]http://www.math.s.chibau.ac.jp/...monster3.jpg[/img]; to get Enjoy ... as you are new here, some posting tips: type [qs]quotes are easy[/qs] and it becomes: and you can type [qs=RAZD]quotes are easy[/qs] and it becomes: RAZD writes: quotes are easy 
or type [quote]quotes are easy[/quote] and it becomes:
quote: quotes are easy
also check out (help) links on any formatting questions when in the reply window. For other formatting tips see Posting Tips For a quick overview see EvC Forum Primer If you have problems with replies see Report Discussion Problems Here 3.0 
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caffeine
Member Posts: 1702 From: Prague, Czech Republic Joined: 10222008 Member Rating: 2.9


Message 6 of 30 (852361)
05092019 1:56 PM

Reply to: Message 3 by AZPaul3 05052019 11:11 PM


First, math is both discovered and invented. The relationship between a circle and its circumference is a discovery. The invented part is the formula for determining what the circumference of a circle is; the symbolic language humans invented to manipulate and make use of the relationship we discovered. 
But I think what the OP might be getting at is that the relationships we're looking at exist only because of how we define the terms. So, we define what a circle is, and can then disCover relationships between it's circumference and area. And we find that these relationships are good approximations to real world objects (only approximations, since a real world circlular object is onlt an approximation to a mathematical circle). But then what about much more esoteric maths? Are quaternions real? You may say their relationships to other mathematical objects are real; and something we discover; but then they only have those relationships because mathematicians defined them into existence by defining the properties of a quaternions. I guess the point is, that we can discover what the logical implications are if we define things to obey certain conditions. And we may find the objects we defined useful for modelling some aspect of the real world. But that does not necessarily mean the real world actually works in any way like the object we've defined  the similarity may break down under certain conditions.
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Theodoric
Member Posts: 6658 From: Northwest, WI, USA Joined: 08152005 Member Rating: 4.2

Great first post
Very nice to see a well though out, well presented first post. Pull up a chair and hang out for a while. Find your niche but participate in any thread that interests you. There is nowhere on the internet where I have to work so hard to make sure I create a coherent, well thought out, well researched post, as here. The old timers will force you to think hard inside and outside of whatever box you bring to the table. It is all meant to make us all more intelligent, more informed and better people. Welcome.
Facts don't lie or have an agenda. Facts are just facts "God did it" is not an argument. It is an excuse for intellectual laziness. If your viewpoint has merits and facts to back it up why would you have to lie?
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Sarah Bellum
Member Posts: 413 Joined: 05042019 Member Rating: 1.1
(1)





Message 8 of 30 (852379)
05092019 6:38 PM

Reply to: Message 5 by RAZD 05092019 9:48 AM


Thanks! I'm happy to find a discussion on mathematics in a creationism/evolution website. From what I've read the topics of evolutionary biology and the Bible have been gone over in great detail (though there's always room for one more!) but not much on math and its interesting philosophical implications (chaos theory, undecidability, nonEuclidean geometry, the Liar Paradox, etc.).
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Sarah Bellum
Member Posts: 413 Joined: 05042019 Member Rating: 1.1


Message 9 of 30 (856408)
06302019 3:58 PM

Reply to: Message 7 by Theodoric 05092019 2:19 PM


Re: Great first post
Thanks!
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Thugpreacha
Member Posts: 12972 From: Denver,Colorado USA Joined: 12302003 Member Rating: 1.2

Re: Great first post
And I just reread it (that first post) and realized it was addressed to me! I think I must have initially ignored and/or overlooked replying because it went over my head. But I will think about it again. I agree with Theodoric. That was a well thought out post, Sarah. Chance as a real force is a myth. It has no basis in reality and no place in scientific inquiry. For science and philosophy to continue to advance in knowledge, chance must be demythologized once and for all. ~RC Sproul "A lie can travel half way around the world while the truth is putting on its shoes." ~Mark Twain " ~" If that's not sufficient for you go soak your head."~Faith You can "get answers" by watching the ducks. That doesn't mean the answers are coming from them.~Ringo Subjectivism may very well undermine Christianity. In the same way that "allowing people to choose what they want to be when they grow up" undermines communism.~Stile
This message is a reply to:   Message 9 by Sarah Bellum, posted 06302019 3:58 PM   Sarah Bellum has acknowledged this reply 

Son Goku
Member Posts: 1152 From: Ireland Joined: 07162005


Message 11 of 30 (856432)
06302019 6:14 PM



Over the years I've gone back and forth on this, but for a while now I've settled more on a combination of 2 and 3 mostly as a result of reading about Godel's theorem and Quantum Theory. The former tying into the whole area of model theory that shows the ambiguity in what any mathematical statement refers to "ontically" and the latter in recent years looking more and more like it points to a nonmathematically modellable layer of reality.

Sarah Bellum
Member Posts: 413 Joined: 05042019 Member Rating: 1.1


Message 12 of 30 (856644)
07012019 11:58 PM

Reply to: Message 11 by Son Goku 06302019 6:14 PM


Can you clarify what you mean when you write, "Godel's theorem ... the whole area of model theory that shows the ambiguity in what any mathematical statement refers to "ontically" "
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Son Goku
Member Posts: 1152 From: Ireland Joined: 07162005
(1)




Model Theory can basically show you that a single mathematical statement can have several models with completely different properties and a single formal statement can have multiple realisations. So let's say the statement "The Reals are not countable". This is a provable statement in the ZFC formalisation of mathematics and basically refers to the fact that there isn't a bijection from the Natural numbers to the Reals. However there are several universes of sets that satisfy the ZFC axioms, these are models of the ZFC axioms. In some models of ZFC the statement "Reals are uncountable" is true because in that model the set that obeys the axioms of the Reals genuinely have a higher cardinality than the set that obeys the axioms of the Naturals. In other models the set filling the role of the Reals actually happens to be of the same cardinality as the set that is the Naturals but a bijection between them doesn't exist. So even a simple statement like "The Reals are uncountable", which seems to say something concrete about the Real numbers, is ambiguous because it's not totally fixed what "Reals" or "Uncountable" refer to. It's a purely formal/linguistic statement in ZFC. Edited by Son Goku, : Slight changes
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Sarah Bellum
Member Posts: 413 Joined: 05042019 Member Rating: 1.1


Message 14 of 30 (856749)
07022019 4:46 PM

Reply to: Message 13 by Son Goku 07022019 2:18 AM


But doesn't the set of real numbers have the cardinality of the set of all subsets of the natural numbers, regardless of what model you're working in?
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Son Goku
Member Posts: 1152 From: Ireland Joined: 07162005

It does. That doesn't change the fact that it might be countable though and thus have the same cardinality from the perspective of a more powerful model. Basically there are models of the reals no larger than the standard model of the Naturals. So even when the Reals have the cardinality of the power set of the Naturals, it can be because they are genuinely larger than the Naturals or they're the same size but the construction of the Power Set is restricted in some way.
