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

ET, often the symbols refer to or even represent and instantiate the facts in question. We have long since shown that the naturals and extensions to C are embedded in any world or in the case of continuum any world that at least conceptually has a space. KFkairosfocus_{December 21, 2018
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hazel:
Pure mathematical facts reside within the symbol system of words and notations we have devised.
Just saying it doesn't make it so.ET_{December 21, 2018
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H, the properties of triangles and circles under stated circumstances have very specific quantitative implications. This reflects the way structure and quantity are embedded in reality; as it is coherent. You composed a specific problem as a part of our study, but that problem rests in turn on the substance. KFkairosfocus_{December 21, 2018
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Here is a scenario to consider, and some thoughts. Let me first say that I am not a philosopher, and don't like playing one in public. But the question of the nature of math is one for which I have practical experience, and some reading background. I have written literally hundreds of pages of math curriculum. Here is a geometry problem that I could have written for a trig class. You don't really need to fully grasp the problem from the description, since it is the philosophy I'm going to discuss, but you might draw a picture if you want to follow along. An isosceles triangle ABC, with vertex at A, is inscribed in a circle. The sides of ABC are 10, and the base is 5. What is the measure of arc AB? Note that especially if I changed the sides to some larger numbers, such as legs of 4356 and a base of 2314, this problem most likely has never existed before in the history of humankind. To me, it seems clear that I invented this problem: in fact, saying I created it might be better. Also, the moment I thought of it, even before I formalized it by writing it up for a worksheet or figured out the answer, I was absolutely sure that one and only one correct answer existed, and that I could easily discover the answer. What can I possible mean that the answer existed before I found it. That is the question I want to think about. (See footnote below for the answer.) If I try to explain this in terms of a Platonic realm, I can't come up with anything reasonable. One possibility is that the Platonic realm has eternally contained every possible mathematical situation, whether someone has thought of it or not: my problem, the Game of Life, the 1 millionth digit of pi in base 17, the status of every point in the complex plane in respect to the Mandelbrot set, etc. For reasons I have explained before, this just doesn't seem like a feasible explanation. Another explanation which occurs to me, but hasn't been mentioned, is that the moment I thought of the problem, it became a part of the Platonic realm, and thus my answer becomes instantly true even before I figure it out. This doesn't seem reasonable. I'm now going to explore another explanation, with no assurance that this will be reasonable either, and with my disclaimer about playing philosopher in mind. Rather than consider ontology, which is what a Platonic realm is about, I would like to just consider epistemology: what we know and how we know it. Logic is inexorable: given some premises within a given mathematical system, assuming the presence of other various theorems, terms, and techniques already established by logic, a whole set of logical conclusions follow. For instance, in my problem, not only could we find arc AB we could find the area inside the circle that is outside the triangle as well. So, in some sense, the potential logical conclusions that could be known are implicitly present in the beginning situation. Just because we haven't articulated the logical steps to reach the answer doesn't mean we can't. Thus, when we say that the answer exists even though we don't know it yet, I think the word "exists" is part of the confusion. It is not an ontological existence that we are talking about. What we are talking about is an epistemological existence: the answer is knowable within the system, but is not yet known to us as an articulated fact. Pure mathematical facts reside within the symbol system of words and notations we have devised. The development of math involves articulating logical consequences of what we already know (including the steps to establish them). We move math facts from unknown to known, which is an epistemological change, but we don't change the fact that they exist within a symbol system: ontologically they remain elements of our logical system whether they are unknown or uknown. =========== Another disclaimer: I know that math can do a marvelous job describing various aspects of our physical world. I don't want to play philosopher on why that is true. I accept it as a fact. But this post has been about pure mathematics, and I want to leave it at that. -------------------------- Footnote: The answer is arc AB = 2 * arc cos (1/4), which I will leave as exercise for the reader. Another solution is arc AB = 180 - arc cos (7/8). This illustrates the point that there can be multiple logical paths to a fact, and that the fact can be expressed in different but logically equivalent ways. Furthermore, I notice that since angle A = arc cos (7/8), arc BC + angle A = 180: they are supplements.hazel_{December 20, 2018
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Here are reasons as to why I believe that mathematics (mathematical truth) is discovered not invented. *1. Numbers have properties that do not appear to have been invented. For example, there some unsolved conjectures about prime numbers that are hard to explain if we are the inventors. Namely if we are the inventors why has no one been able to prove (or disprove) that the set of twin primes is infinite? Or why do the Goldbach conjecture and Riemann Hypothesis continue to be unsolved? Wouldn’t the putative inventors of mathematics be able to resolve these problems? *2. The applicability of mathematics to the physical world. For example, earlier in this thread I pointed out that “One of the most significant discoveries in science was the discovery of the inverse square law (credited to Kepler for light) which is derived directly from the geometry of a sphere. The ISL applies to both electromagnetism and gravity, though the force constants for each vary.” https://www.thehighersidechatsplus.com/forums/media/inverse-square-law-and-wave-function.105/full?d=1503980290 Where would physics be without this discovery? And that’s only one example. *3. It appears that the human mind and brain are preadapted to do mathematics. What survival advantage would doing math and doing it accurately have for a highly evolved species of hunter-gatherer apes? *4. The universality of mathematics. SETI enthusiasts have suggested that we could use mathematics to communicate with ETI’s. For example, “In the 1985 science fiction novel Contact, Carl Sagan explored in some depth how a message might be constructed to allow communication with an alien civilization, using prime numbers as a starting point, followed by various universal principles and facts of mathematics and science.” https://en.wikipedia.org/wiki/Communication_with_extraterrestrial_intelligence How could mathematics be universal if it was invented by us? *5. Historically mathematics set the stage for the scientific revolution. Kepler and Galileo and Newton were all mathematicians who believed that at its root the universe was mathematical. In other words, they began with the assumption that the universe could be described mathematically. Does anyone have anything to add?john_a_designer_{December 20, 2018
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H, note the summary from Needham in 211. Notice how properties of circular motion show how complex numbers act as vectors. Similarly, note how the exponential function emerges as a characteristic function under the operation D*. Onward, a lot of geometry comes out, including a whole perspective rooted in translations and reflections. More coherence and intelligibility of rational principles of structure and quantity. KFkairosfocus_{December 20, 2018
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P.S. I'm having some new thoughts that clarify things a bit for myself: hope to post them sometime today.hazel_{December 20, 2018
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I don't think so. I have also discussed reasons why I don't acknowledge platonic reality. The fact that there are logical conclusions that must be true even if we never explicitly follow the steps that lead to them says something about the nature of logical implication, but not necessarily anything about ontology. Reminder: I am of two minds (or more) about all this, but I can't agree that, based on one of the things I said, without taking others into account, that I acknowledge platonic reality. The things I said about about what I can't accept about the idea of platonic reality are deal-breakers for me. YMMV.hazel_{December 20, 2018
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hazel: I agree and have discussed above that one of the features of math is that the things we discover are true whether we have discovered them or not. If you what you're saying is that implications in our abstract maths are true whether they have been perceived by us or not, this seems to be an acknowledgement of the platonic reality. If you ponder this statement in a meditative state, you may experience a koan. ;)mike1962_{December 20, 2018
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Ed George:
but is this a case of discovering the mathematics inherent in the universe, or finding applications for the mathematics we invented.
You have yet to make a case that we invented anything.ET_{December 20, 2018
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PS: Tristam Needham develops thoughts at book length, note responses: http://usf.usfca.edu/vca/ and raises profound issues. He highlights that Newton employed two distinct approaches to Fluxions (aka Calculus), a power series one AND a later geometric one, as opposed to algebraic formalisms that sometimes help [and seem to be "more rigorous"] but which tend to lock out the visual-spatial aspects of insight into structure and quantity. For example, he applies the i-operator approach, asking what happens when on a continuous basis velocity -- trajectory in an ideal space -- is at right angles to position vector? This is of course bringing to bear the i* operation, and allows us to recall that this v- is- ever- tangential- to- z . . . the complex vector . . . is a key defining aspect of the kinematics of circular motion. At (1,0) of course velocity is directly upward, at (0,1) it will be leftward, and at (-1,0) it will be downward. In between, we will ever be pointing v in the leftward (anticlockwise) tangent direction to z considered from the origin, the pivot of the rotation. We thus see that, naturally, i* is about rotation and here circular motion. We can of course readily apply the orthogonal components of z, x --> r cos wt and y --> r sin wt. The three power series expansions dovetail once we use i* y, giving the link to z = r e^iwt. He makes some historical observations:
far from being embraced, complex numbers [which appeared -- with suspicion -- in 1545] were initially greeted with suspicion, confusion, and even hostility . . . . The root cause of all this trouble seems to have been a psychological or philo-sophical block. How could one investigate these matters with enthusiasm or confi-dence when nobody felt they knew the answer to the question, “What is a complex number?” A satisfactory answer to this question was only found at the end of the eigh-teenth century z. Independently, and in rapid succession, Wessel, Argand, and Gauss all recognized that complex numbers could be given a simple, concrete, geometric interpretation as points (or vectors) in the plane: The mystical quantity a + i b should be viewed simply as the point in the xy-plane having Cartesian coordinates (a, b), or equivalently as the vector connecting the origin to that point. See [1]. When thought of in this way, the plane is denoted (C and is called the complex plane3.
Sounds familiar? He continues, applying the vector approach:
The sum A +B of two complex numbers is given by the parallelogram (1) rule of ordinary vector addition. The length of AB is the product of the lengths of A and B, and the 2 angle of AB is the sum of the angles of A and B [from the polar axis, ox].
Back on the worldviews front:
The publication of the geometric interpretation by Wessel and by Argand went all but unnoticed, but the reputation of Gauss (as great then as it is now) ensured wide dissemination and acceptance of complex numbers as points in the plane. Perhaps less important than the details of this new interpretation (at least initially) was the mere fact that there now existed some way of making sense of these numbers——that they were now legitimate objects of investigation. In any event, the ?oodgates of invention were about to open.
The name for this, is paradigm shift, where endorsement by a star of the field (yes, academia has long had a celebrity culture) triggered an acceptance cascade . . . shifting the overton window for Mathematics. But 200 years later, too much of education on the subject has not caught up. (And the discussion on quadratics and cubics is useful too.) Another key insight (I substitute wt for theta and vt for phi and use / for the angle notation that looks like a capital L bent into acute angle shape):
Let z denote a general point [= position vector relative to origin and polar axis] in C, and consider what happens to it— where it moves to-—when it is multiplied by a fixed complex number A = R/vt. According to [the vector product rule], the length of z is magnified by R, while the angle of z is increased by vt. Now imagine that this is done simultaneously to every point of the plane: Geometrically, multiplication [of the plane of z's] by a complex number A = R/vt is a rotation of the plane through angle vt, and an expansion of the plane (9) by factor R.
Thinking spatially is already opening up new vistas. To do so, he brings in Euler's e^iwt = cos wt + i sin wt. This can be seen as a vector sum on components and can be expressed as power series expansions. He goes on:
Instead of writing a general complex number as z = r / wt, we can now write z = r e^ i*wt. Concretely, this says that to reach z we must take the unit vector e^i*wt that points at z, then stretch it by the length of z.
Going on, and using D* as derivative operator d[]/dx:
Recall the basic fact that e^x is its own derivative: D*e^x = e^x. This is actually a defining property, that is, if D* f(x) = f(x), and f(0) = 1, then f(x) = e^x. [--> characteristic function under the operation D*] Similarly, if k is a real constant, then e^kx may be defined by the property D*f(x) = k f(x). To extend the action of the ordinary exponential function e^x from real values of x to imaginary ones, let us cling to this property by insisting that it remain true if k = i, so that D* e^it = i*e^it . (11) We have used the letter t instead of x because we will now think of the variable as being time. [--> we have been there all along, think of positions (x,y) and trajectories manifested in velocities etc]
We already can apply that i* means A/C rot through a right angle and we see that we have rate of change of position here as perpendicular to position vector at all times. That is, naturally, we are looking at circular, vector motion in the plane. Following Needham:
. . . velocity = V = iZ = position, rotated through a right angle. Since the initial position of the particle is Z(0) = e^0 = 1, its initial velocity is i [upwards], and so it is moving vertically upwards. A split second later the particle will have moved very slightly in this direction, and its new velocity will be at right angles to its new position vector. Continuing to construct the motion in this way, it is clear that the particle will travel round the unit circle.
In short, we need multiple perspectives, which mutually reinforce. And here we see how structures and quantities tied to space [which requires 2-d continuum] allow us to see how complex numbers understood vectorially have a very natural interpretation. One, embedded in the realities of space. And note how rates and accumulations of change across time in space -- trajectories -- are also deeply involved.kairosfocus_{December 20, 2018
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F/N2: This discussion brings in cross-perspectives on defining exponential functions (in effect what is the characteristic function of differentiation?) and on the import of vector based trajectories: https://books.google.ms/books?id=ogz5FjmiqlQC&lpg=PP1&pg=PA10&redir_esc=y&hl=en#v=onepage&q&f=false KFkairosfocus_{December 20, 2018
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F/N: A useful video explanation of the Euler expression: https://www.youtube.com/watch?time_continue=24&v=qpOj98VNJi4 KFkairosfocus_{December 20, 2018
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MG, very well noted. In addition, there is elegant simplicity that embeds the profound . . . and thus excites the aesthetic/ axiological wonder triggered by beauty. In electronics, I once sat in a workshop by a genius circuit designer, who used a classic fixed bias transistor ckt in subtle ways. A lesson. Euler's expression as I discussed last evening is similar. KFkairosfocus_{December 20, 2018
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WJM @ 161 "I think the question now becomes: what can we infer from the fact that mathematics is an inherent aspect of the universe and of our existence?" First off, let's review the obvious fact that A-mat is incompatible with an infinite, eternal (mathematical) platonic realm. Such a realm requires intelligence (perhaps among other traits) to access. Aspects of the realm require genius to perceive (which is why my own access is rather minimal :-) We see some highly sophisticated mathematics that undergirds our physical universe, in particular QM which was discovered by genius minds such as Planck, Bohr, Einstein, Schroedinger, Heisenberg, Dirac, Feynman, et al. As BA77 is wont to extol, QM is full of counterintuitive facts. Even the great Einstein did not want to accept quantum entanglement, but it is now an established fact. Continuing by analogy, there should not be an upper bound on the complexity of relationships and laws in the platonic realm (in fact, Godel's theorems imply this). Some of the relationships/laws will be beyond human reach and only be accessible to godlike intelligence. I suspect that subtle undiscovered laws governing our own universe belong to that class.math guy_{December 19, 2018
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MG@24, but is this a case of discovering the mathematics inherent in the universe, or finding applications for the mathematics we invented. In your examples I would suggest that it is the latter.Ed George_{December 19, 2018
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I totally agree, MG. I'm pretty sure I said somewhere previously in this thread that a lot of math has first been developed as pure math, and then found to have application to the real world. Complex numbers, which we have discussed a lot here, are one such topic.hazel_{December 19, 2018
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h @ 201 "And lots of math has been logically discovered that doesn’t apply to any model of any aspect of the physical world" A little humility might be in order here. I would insert YET into that sentence. For example, more than 70 years ago G.H Hardy wrote to the effect that his beloved number theory was pure and unadulterated by military applications (actually any application outside pure math) unlike calculus, say. But his sentiment was premature since number theory underlies modern cryptography which is crucial for military signal processing (and of course e-commerce). Algebraic topology was another "pure math" discipline with little use in the real world. But now persistent homology is an important tool for network analysis and finding patterns in huge data sets.math guy_{December 19, 2018
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That's a nice paper. His derivation of Euler's formula is exactly like I used to teach it in calculus. As a small point, and I think he makes this mistake, Euler's formula is e^(ix) = cos x + i sin x, and Euler's Identity is then e^(i•pi) = -1. The first is a formula for any x, and the second is just a fact, like sin 30 = 1/2.hazel_{December 19, 2018
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PS: I found an interesting first level discussion: http://travismallett.com/wp-content/uploads/2013/02/MATH-182-Euler_Article.pdfkairosfocus_{December 19, 2018
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re 199: Yes. it was my opinion also that "Euler’s Identity itself does not model any physical phenomenon." I don't know whether you've read all the intervening posts, but I've discussed how teaching complex numbers was one of my favorite subjects, and kf and I have both remarked on how useful they are in described some important and complex phenomena. You ask, "Given the nature of the universe, were humans bound to create a math system where EI is true? ... Our best math that contains (as far as we can tell) the essential terms pi, e, i, somehow managed to have such a nature that EI is true. Moreover, the math implied EI before Euler discovered it." First, I agree and have discussed above that one of the features of math is that the things we discover are true whether we have discovered them or not. I'm not sure we can say that humans were "bound to" discover EI, because the history of math, and human civilization in general, might not have gotten there. (For instance, there may be wonderful mathematical topics, tools, and results that we have not developed and discovered yet, and might not ever.) But given making the basic decisions about terms and the work we have done, the EI is a straightforward result of starting with the number system and developing trig, exponentials, and complex numbers. At 70, I wrote,
A common distinction is that mankind has invented the particular symbol systems that we use, but within those systems, once established, the logical consequences are then discovered as inevitable logical consequences. For instance, once the number system was extended to include complex numbers, Euler’s Identity e^(i*pi)= -1 was discovered.
and at 170
As I used to tell my seniors, they walked in to first grade learning the number fact 1 + 1 = 2 and walked out knowing the number fact e^(i*pi) = -1. These facts, and everything in between, given suitable definitions, are logical consequences, contained within the system, that go back to the original structure of the natural numbers.
So as far as all this goes, I think we are in agreement. You write,
I assume that the universe pre-existed humans, and I assume that it has actual properties that can be perceived, so to the degree that our maths describe the real world, the maths have been discovered. This seems obvious. What am I missing?
I think there are two different kinds of things that we can say have been discovered. As a purely mathematical fact, we discovered that e^(i*pi) = -1 as a logical conclusion based on the foundational number system and suitable definitions we developed along the way. This is a logical discovery. However, the fact that complex numbers can be used to model the quantum wave function, for instance, is an empirical discovery: people had to propose a model using complex numbers and test it against empirical evidence to see if it worked. And lots of math has been logically discovered that doesn't apply to any model of any aspect of the physical world, so there has been no corresponding discovery of a workable description of some part of the world that uses that math.hazel_{December 19, 2018
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M62, It is obvious that the five numbers in the Euler expression are ubiquitous in many contexts of pure and applied mathematics, the latter engaging the real world in many practical ways. What is perhaps most interesting is that they are exceedingly diverse: 1 and 0 are foundational to the natural numbers, which are indeed core to Math. The circle's geometry defines pi, which is another ancient branch of Mathematics. Then, e and i are at the heart of modern developments and applications since the 1600's, again a very divergent situation. However, the complex plane and complex exponential forms of complex numbers [which are tied to sinusoids and to power series representations) do point to a unit length vector sweeping the unit circle in the complex plane. In that context we may ponder z = 1* [e^i*w*t] and the case where wt = pi rads, this notoriously being "the natural unit of angle." (Yes, even units of angle may not be arbitrary!) Once we see this and re-arrange, we get the astonishing unification: 0 = 1 + e^i*pi. This means that the five key numbers and key operations: sum, product, exponentiation are locked together to literally infinite precision, and that this will apply in any world where at least a 2-d conceptual space is possible (which requires the continuum, thus N --> Z --> Q --> R as sets, then going vector through i*R an orthogonal transform of R, thus C). One take-home is that here we have reason to be confident in the coherence of a wide sweep of Mathematics, something not to be taken for granted post Godel. And this includes the power of transforms to bring differential equations to the complex plane, then by extension difference equations -- instantly applicable to a lot of abstract and applied system dynamics. There is much more, those who would trivialise the result would be well advised to reconsider. Where, of course, all of this is pointing to the way the substance of logical principles of structure and quantity are embedded in possible worlds . . . one of the themes it seems some struggle to accept as material and significant. KFkairosfocus_{December 19, 2018
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Hazel: 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. Euler's Identity itself does not model any physical phenomenon. But the terms are broadly employed in physical models, and integral to our best and wildly accurate models, such as quantum physics and General Relativity. Our known perceptions and discoveries of the known physical world certainly seem to indicate that pi, e, and i are fundamental in describing the universe's properties. This is not controversial. Maybe our accuracy is off a bit, but we're in the ballpark, no? What is interesting about EI is that these ubiquitously applied, physically accurate terms have such a "simple", elegant, fundamental relationship. Given the nature of the universe, were humans bound to create a math system where EI is true? If not, what a lucky accident. Our best math that contains (as far as we can tell) the essential terms pi, e, i, somehow managed to have such a nature that EI is true. Moreover, the math implied EI before Euler discovered it. Any mathematical or symbolic system has implications that are true whether anyone has perceived them or not. In that sense, anything not fundamental is discovered and not invented. I assume that the universe pre-existed humans, and I assume that it has actual properties that can be perceived, so to the degree that our maths describe the real world, the maths have been discovered. This seems obvious. What am I missing?mike1962_{December 19, 2018
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To JAD and others: John Conways game of life and wikipedia's article Lizzie/ febble/ queen penguin used to love that one, too, hazel. I am still not sure how it is relevant to the claim that mathematics permeates the universe because mathematics was used to intelligently design it. Does it really bother people that the intelligent humans merely discovered, rather than invented, these amazing gifts they have given us? To me their ability to tap into the universal information and make sense of it is inspirational.ET_{December 19, 2018
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H, I will continue to point out what is plainly well warranted and highly material, despite the all too common tendency to sideline or ignore or even sometimes to try to make such seem false or dubious or to exert inappropriate selective hyperskepticism to dismiss. I will also continue to show that such reflects the balance on the merits. KFkairosfocus_{December 19, 2018
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kf writes, "H, I repeat" Yes, you do. :-)hazel_{December 19, 2018
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H, I repeat, there is a logic of structure and quantity that pervades our observed cosmos and in part such will pervade any possible world. Our study of the logic of structure and quantity and abstract model worlds we may create in that study in the end will have to be constrained by that substance. For instance, many times now I have shown that distinct identity -- a necessity for any particular world -- directly has the natural numbers as a corollary. We use fingers, sticks, hash-marks etc in simple Mathematics precisely because of the principle that two sets with the same cardinality can be put in one to one correspondence. In short, our discovery of numbers is tied to the embedded reality of same in any possible world. Where too, I have pointed out that more sophisticated abstract logic model worlds are possible worlds, so entities we discover in exploring them which are necessary entities will also be present in all possible worlds. Similarly, if the logic model is sufficiently similar in material respects, some things in that model will be in the world we inhabit. The utility of Mathematics traces to these factors. KFkairosfocus_{December 19, 2018
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JAD, I have emphasised the contrast between the substance of the logic of structure and quantity and our study of it. KFkairosfocus_{December 19, 2018
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EG, your comment was on a series of OP's I have been developing on logic and first principles. KFkairosfocus_{December 19, 2018
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JAD writes, "So what the issue now with games? I agree games are something we invent, and mathematicians have long been fascinated with them. So?" The Game of Life is not really a game. It's an iterative model that incorporates, at a simpler level, the same principles that produces fractals such as the Mandelbrot set. It's a mathematical system. If you're not familiar with it, I suggest that you find out some about it.hazel_{December 19, 2018
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