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| Infinity, does it extent both inwards and outwards ?
Posted: 10/31/2009 2:47:12 PM | | Buddhism? I thought I was talking about elementary logic??...Oh well, be that as it may, I have to take issue with you on the matter of infinity, because (for instance) if the "big bang" is an "open" universe, there will be no "end of time" and spacetime will expand forever. Gee...sounds like an infinity to me! | |
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| Infinity, does it extent both inwards and outwards ?
Posted: 10/31/2009 3:49:27 PM | RE Msg: 73 by lsdime:
If you stop to think about it, the word INFINITY has alot of Newtonian connotations. Our universe is much stranger than newtonian physics. In our universe, the word infinity, as most people think of it, does not exist. Sorry, but that's just a whole load of hooey. Anyone who has studied first-year analysis knows that. Infinity has a very specific definition, that covers things that could never fit into a Newtonian universe, like Peano shapes, and Lebesgue integrals. Infinity, as most people think of it, DOES exist. But, what most people DON'T know, is that beyond infinity, is ANOTHER infinity, and ANOTHER, and ANOTHER, and that allows infinity to go outwards, inwards, and billions of ways.
But I doubt that's really delved into too deeply in M-theory, because M-theory is treated as theoretical physics, and the nature of infinity is covered in pure maths, and physicists tend to avoid pure maths like the plague. | |
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| Infinity, does it extent both inwards and outwards ?
Posted: 10/31/2009 4:26:03 PM | Isdime,
thank you.
so people are confusing Space and our Universe when thinking on Infinity?
our universe has boundaries , Universe = finite , but Space- when proven- may not, Space= > x ?
Dale | |
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| Infinity, does it extent both inwards and outwards ?
Posted: 10/31/2009 4:35:01 PM | Yes infinity does exist both directions.
The better microscopes become, the smaller life that is found.
As for the bigger side of things - well there is much beyond our universe, and our universe is rather young still. Metephorically speeking, I'd say we are still in the womb.
And looking at some of the hubble images of the universes brain, anatomy and other features like the hour glass eye - it's quite remarkable how similuar the anatomy of the universe is to the anatomy of us humans! Of course we are made of much simpler material (flesh and bone).
Life goes on to infinity, it always has. The life cycle, not the life line!!!
An entire cycle of life was the beginning, life didn't start at the same kind of beginning most science is searching for - most science is still looking for a straight line rather than just accepting life!
The more answers science finds, the more questions there will always be.
Don't get me wrong, our physical universe is important to our physical being - but we shouldn't get hung up on it too much - science and spirituality should be treated equally, and the physical realities should not be more important then the spirituality many have lost touch with. They are 2 opposite side of the same thing. | |
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| Infinity, does it extent both inwards and outwards ?
Posted: 10/31/2009 4:59:52 PM | Sorry, but that's just a whole load of hooey. Anyone who has studied first-year analysis knows that. Infinity has a very specific definition, that covers things that could never fit into a Newtonian universe, like Peano shapes, and Lebesgue integrals. Infinity, as most people think of it, DOES exist. But, what most people DON'T know, is that beyond infinity, is ANOTHER infinity, and ANOTHER, and ANOTHER, and that allows infinity to go outwards, inwards, and billions of ways.
wow, speaking of loads of hooey. You seem pretty sure about yourself scorp. Do you posses some knowledge that the world's leading cosmologists are unaware of? You claim that MOST people don't know is that there are more infinities beyond our universe....actually, NOBODY knows that, because its just spectulation. This paragraph is nothing but faith....religion for all practical purposes. Pardon me, scorp, but i will stick to what we actually know, or can know, through experiment. And what we know so far is that our universe has an event horizon...and likely beyond that we would track space back to the big bang..(according to relativity). Anything you say beyond that is spectulation...or should i say...."hooey". But thanks for sharing your personal faith-based belief.
oh, by the way, Peano shapes, and Lebesgue integrals are nothing but mathematical constructs. There has never been anything showing that they actually apply to anything in our physical universe. You'd think a first year "analysis" student would know that. :)
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| Infinity, does it extent both inwards and outwards ?
Posted: 10/31/2009 5:55:49 PM | Personally I like the theory of fractal inward and vortex expansion.
Allowing for infinity in both directions.
Perhaps the limits or finites we experience, are just confounds of our dimension.
A truly beautiful model can be seen when you consider a simple triangle –
To understand inward fractal you overlap one triangle with another,--half way.
It looks like the Star of David—6-point star
Now look at the triangles as a tetrahedron 3 dimensions—and you have a fractal of smaller triangles.
The vortex can be modeled using an expanding circular spin outwards
Infinitely big--- infinitely small | |
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| Infinity, does it extent both inwards and outwards ?
Posted: 10/31/2009 6:57:23 PM | That is true...
Pure mathematics deals with abstract things that has nothing to do with the Universe or Anything Physical. At least thats what we thought. Many of the abstract 20th century mathematics which we believed to be far removed from anything physical ended up being used for various different concepts in String Theory and M-Theory. Now many of the string theorist and Mathematical physicists are delving into pure mathematics. Some pure Mathematics came out of string theory believe it or not. Edward Witten one of the leaders in string theory is said to be rivaled by few Mathematicians. And as far as sting theory goes if the equations are to be completely understood then we need smarter people and new Mathematics. Is that surprising!
As far as the Universe goes we can't quiet be sure if there is an outside space which exists independently from our universe. And since space is possibly flat then we usually think of space as expanding outwards from the big bang's singularity or plank length. So since we can never travel outside the universe or reach its edge since space is or will soon be expanding faster then light. The question then becomes Unanswerable.
I believe Infinities only exists in Pure Mathematics as Pure Abstract Concepts. And even if there existed infinities in the real world(Universe) we can never truly understand its true being and therefore it becomes unknowable. We can also say its essence can't be known by finite or even infinite beings.
Physics deals with infinities everyday. When they do their calculations each day they must get finite results or throw away it away. So physics can be said to get finite results from infinite different variables. Take for instance when Renormalization was created from quantum electrodynamics it was to make sense of infinite integrals in perturbation theory. So they used infinities to cancel out infinities if that makes sense. Like Albert Einstein said, "as far as the laws of mathematics refer to reality, they are not certain; and as far as they are certain, they do not refer to reality."
So the jury is still out on physical infinities... | |
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| Infinity, does it extent both inwards and outwards ?
Posted: 10/31/2009 9:01:32 PM | @ Nate
Good post! As a burned-out old hippie from the sixties, I have to say that the philosophy of mathematics is my "thing", so I enjoy it on those rare occasions that a thread pops up discussing it.
I believe Infinities only exists in Pure Mathematics as Pure Abstract Concepts.
I believe mathematics itself exists as a pure abstract concept, like a language, so let's explore the "mathematics as language" concept for a moment and see how it might relate to "infinity":
Let's take our concept of language in general and see where it applies. Now there are a lot of mathematicians one might call "Platonists". They would insist all mathematics exists in some analogy of Plato's world of forms and only awaits "discovery." The mathematical formalists (most mathematicians) would insist as I do that mathematics is a language, merely a collection of symbols connected by rules of "grammar" to create mathematical expressions. They would further assert (as I no longer do) that it is just a game they play and any correspondence with what we call "reality" is only the most astonishing of coincidences. Then we have the constructivists, who claim that the only mathematics that exists is the mathematics we can define and express without fear of contradiction. Basically, they don't allow things like "infinity" to exist because of the mathematical problems it creates. The "problem" created by the constructivists is the renunciation of much of modern mathematics. There is a fear running though the mathematical community that the costructivists are "terrorists" who, if they had their way, would throw the baby out with the bathwater, just to make sure there are no holes in the tub.
If we are looking at mathematics as language, let's compare it to English. English is a fairly expressive language, but it does have many inconsistencies and allows words to be strung together in such a way that makes no sense. For instance, I could say "I don't know ware I was last night." In writing, it makes no logical sense, and one might strongly suspect I screwed it up somewhere. If it was only spoken, however, it would sound perfectly fine to you because "ware" sounds enough like "where" to pass for it audibly and make the sentence seem correct. Should we reduce the language's ambiguities by placing on it the condition that no two words can be allowed to sound alike? (a Herculean task requiring a complete reformulation of english), or should we just try to live with it and remove ambiguities as we find them? It would seem more reasonable to follow the latter course, but we must be aware that there will continue to be much nonsense said by those using the language that will slip past even the "literati." So it is with mathematics. Unless the tools used are well known not to produce ambiguities, there is no guarantee that inconsistency will never raise its ugly head and even non-constructive formal proofs may have to be taken with a grain of salt.
So we shouldn't be too critical of the constructivists for the valuable work they do in trying to build a consistent mathematics. I have great sympathy for them. I'm not quite ready to renounce the law of the excluded middle yet (much as I would like to exclude my rather prominent middle from discussion), finding it as useful as I do, but I do bear in mind when I use it that I could be "riding for a fall."
The next question is one of existence itself. If something doesn't exist as a "physical" part of the universe, does it not exist at all? What about English? Does it have a physical existence in our universe? If so, where is it? If not, what are we using when we communicate? English is an abstract structure without physical existence, pretty much the same as mathematics. Both languages help us communicate, describe and derive much to say about the universe that surrounds us. When we string a few rules or axioms together, we create a structure in a fog. We know it's there, but must explore it to see what it is like. Can anyone see all of arithmetic on looking at its axioms? Why can we not then say that mathematics (or the structure of it, or better yet, the potential of structure) exists in some non-physical sense? If we do, why don't we call it a "world of forms" and make Plato happy? If we do, can we not see that the battle between the Platonists, formalists and constructivists is kind of a tempest in a teapot and that they are all sort of right? (should this principle of unification be applied to party politics?)
Now I'm ready to discuss infinity...Well, maybe not. This post is getting too long. Maybe I should just say for now that infinity exists in Plato's world of forms, the formalists work with it and the constructivists won't allow it. Does it "really" exist in our physical universe? How the hell should I know??, but I believe it does if I'm willing to allow it. | |
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| Infinity, does it extent both inwards and outwards ?
Posted: 11/1/2009 5:42:53 AM | Dukky,
so it's sort of like using the Gestalt Theory, that the formalists,constructivists, Platonists are trying to unite behind?
in terms of all beliefs of these three are trying to congeal. they maybe should use an overall basis on things that they can agree on, and for those which they can not, they should see if there's a part of any of the 3 that works and use it for such an occasion.
each disagreement settled by what works best for the problem( without their pride in the way) and use it as an answer to the problem.
sort of making the sword sharper by hammering it out in the forge when it is being formed, and then not have to sharpen it as much with a whetstone later on.
would this point of view work, if the "we know better" persons of the 3 partitions stand silent long enough to actually learn something?
Dale | |
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| Infinity, does it extent both inwards and outwards ?
Posted: 11/1/2009 9:34:41 AM | RE Msg: 80 by lsdime:
You seem pretty sure about yourself scorp. Do you posses some knowledge that the world's leading cosmologists are unaware of? No. I've just grown up with the mathematical and scientific developments of the last 100 years, and taken them on board. Scientific knowledge has evolved. It's now a question of which of our species adapts to these new discoveries, and which don't. I've adapted.
You claim that MOST people don't know is that there are more infinities beyond our universe....actually, NOBODY knows that, because its just spectulation. This paragraph is nothing but faith....religion for all practical purposes. It's speculation that infinity exists as well. It's speculation to say what happened in the past, or the future. It's speculation of what happens outside our solar system. But, science is founded through rigorous analysis of such speculations, and that is exactly what multiple infinities are all about, rigorous analysis.
Pardon me, scorp, but i will stick to what we actually know, or can know, through experiment. And what we know so far is that our universe has an event horizon...and likely beyond that we would track space back to the big bang..(according to relativity). Anything you say beyond that is spectulation...or should i say...."hooey". But thanks for sharing your personal faith-based belief. I can agree that Newtonian determinism and empiricism has taught us a lot over the 17th, 18th and 19th centuries, primarily because we weren't considering them before that. However, the late 19th and 20th centuries have shown us that empiricism is incorrect, and that determinism is faulty as well. We've seen these developments, in philosophy, in mathematics, and in Physics. It only remains to embrace them, and evolve science past the beliefs of the 16th century.
However, it is scary to many people, as it requires a change of view, and a change in how we view the world, which necessarily questions if the things we trust in are truly reliable. So it's perfectly understandable that you'd want to stick to the past.
But really, it's not all that scary, provided we are willing to consider into what it means, as where it leads us, is to a change of view, that makes us realise the world is more secure than it appears to be, just not one that works with previous views of the universe.
oh, by the way, Peano shapes, and Lebesgue integrals are nothing but mathematical constructs. There has never been anything showing that they actually apply to anything in our physical universe. You'd think a first year "analysis" student would know that. :) Lebesgue integrals are interesting to study, not just as a construct, but because they allow us to integrate many situations in which normal Newtonian-Leibnizian calculus doesn't work. Traditional calculus works best on 1-st order linear equations over a uniform space. Trouble is, the universe isn't often linear, and it is isn't often uniform. We used to fit "square pegs" into "round holes", by just approximating equations to linear forms, and approximating non-uniform spaces to uniform spaces. But the more we are investigating the universe, the more we find that just doesn't work.
Lebesgue integrals are used to generate Lebesgue spaces, which include standard probability spaces. It turns out that any probability distribution using real numbers, or using multiple dimensions of real numbers, is a standard probability space. That makes them extremely useful for any situation relying heavily on probability, where things aren't uniformly linear, such as quantum mechanics. But, it also applies equally well to any scientific study that uses probability to deduce its results, which is almost every scientific study nowadays, and where normal linear uniform methods just don't work well, which is probably most current scientific studies. I expect that it would be very useful to use Lebesgue integrals in everything from studies in evolutionary processes, to predicting super-waves along shipping lanes, to building more efficient semiconductors, because even electron current isn't always uniform.
All such ideas are very useful. But it is true to say that they go beyond the 16th century need for uniformity and linear thinking. But then, we're way past that as well.
On topic, all proofs and theorems of mathematics are things that can be explained equally well in English, to a layman. If not, they really aren't part of mathematics, because maths just uses symbols. One of my lecturers gave his entire course in English, with hardly any symbols at all. But it was a lot more wordy, and a lot less clear, and that's why mathematicians like using symbols. Symbols let you say something in very few lines, and in a way that can be proved definitely true or false. Sure, they're abstract, about as abstract as any discussion that anyone has. | |
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| Infinity, does it extent both inwards and outwards ?
Posted: 11/1/2009 9:54:44 AM | @ Dale
Actually, the history of the "schools" of Platonism, formalism and constructivism is quite interesting. A little reading about their history might be in order (to get a better "feel" for them). But basically, mathematics started out as "Platonist" in the sense that it was considered divine, or a gift from the Gods that Man could work with. Plato gave this "divine" theory of mathematics a sort of philosophical foundation, and for centuries it was assumed he was right. Only with the repudiation of "forms" in many schools of philosophy (developed mostly in the period of the Enlightenment) did it become necessary to find another "source" for mathematics. The formalist school grew from this (in denying "forms" it denied a divine origin for mathematics, but it still retained some assumptions that can only be termed "faith-based"). The constructivists came along later, when some mathematicians wanted to purge mathematics of inconsistency and the things that allowed it. (i.e. It would no longer be sufficient to prove something based on things like the law of the excluded middle (something that would be worth looking up for a better understanding of the constructivist position))
It has been said that a mathematician is a Platonist at heart and really believes mathematical objects exist and are meaningful, but when asked, he'll hedge and say he's a formalist and mathematics is just a meaningless game of symbols & rules that he plays for a living (though it will usually stymie him that math applies so well to the real world). The constructivist (or "intuitionist") might say that mathematics is an edifice "in progress" and doesn't exist until he lays the stones (and they have to be on a solid foundation of previously laid stones). He disdains the formalist's assumption that a complete mathematical structure is defined by the existence of a foundation and would say that the only structure that exists is that created by the bricks that are laid in place. Only when it is built will we know how the structure should look (and of course it is unlikely to ever be finished).
Maybe the best way to get a "feel" for the differences in mathematical philosophy is to look up some of the definitions. I avoided using the word "intuitionist" previously, because of MAJOR differences between the schools on what intuition is and the confusion it would create in most minds to call a constructivist an intuitionist. Bearing that in mind, we should perhaps explore philosophical "intuitionism" (I hate "isms", but sometimes it is necessary to explore them). There seems to be a pretty accurate article on Wikipedia about it:
http://en.wikipedia.org/wiki/Intuitionism
As an aside, I have to say it has always made me feel quite "God-like" whenever I said "Let there exist..."  | |
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| Infinity, does it extent both inwards and outwards ?
Posted: 11/1/2009 10:19:55 AM |
all proofs and theorems of mathematics are things that can be explained equally well in English
I'd say that is a false assumption on your part, Scorp, unless you can formally prove it true for the whole set of theorems. (sorry -- it's the constructivist in me.) | |
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| Infinity, does it extent both inwards and outwards ?
Posted: 11/1/2009 10:59:23 AM | RE Msg: 87 by JustDukky:
all proofs and theorems of mathematics are things that can be explained equally well in English I'd say that is a false assumption on your part, Scorp, unless you can formally prove it true for the whole set of theorems.
(sorry -- it's the constructivist in me.) I'm not against constructivism, as that's pretty much how I learn things, by constructing new ideas based on what I already know.
Constructionism doesn't always work, as sometimes I find that the particular constructions I was taught are so full of contradictions and counter-examples, that I have to abandon them. But then I almost always find that another construction does explain the idea. So in my view, the problems with constructionism are not that it doesn't always work, but that usually when it does that the choice of construction is not always a good choice, and that's more human error in choosing the wrong construction than anything else.
My view is that when I was in university, a lecturer once told us that "mathematics is a language, with vocabulary and grammar". This made a huge impression on me, as I had never heard the idea before. But upon reflection, I realised that everything that I was taught in mathematics, from substitution, to the definition of a Hahn-Banach Space, was explained using English, or using symbols and English, and those symbols had been previously explained using English, or using symbols and English, which in turn were explained using English, or symbols and English, and that the most primitive symbols that I had, were all originally explained to me, using English. If not for that, I would not have been able to understand the formulas and theorems that used those symbols.
This was further confirmed to me, when during one lecture, I realised that when we were being taught a particular substitution as a step of the proof that we were studying, that we weren't being taught any symbolism at all to express the substitution, but were explained how the substitution worked, using basic English. I realised that I would have great difficulty in explaining how this substitution would work in mathematical terms, and could only explain it in English, and so would all the other students in my class. It then hit me that most proofs I'd learned had used exactly the same methods, steps that were explained in English, but not explained in mathematical terms. It therefore hit me that were we to try to fully mathematicise all proofs, that we'd have a massive problem in doing so, because most proofs of mathematical theorems are taught in exactly the same way.
This was further confirmed to me by one lecturer, who chose to teach all of that course using almost nothing but English. But doing revision, it was way too wordy, and it wasn't very clear to me. So I had to re-write his course to do revision on it, and in the end, I condensed over 30 pages of English, into 2 pages of mathematical symbols. It was then that I realised that mathematical symbolism is just used to condense reasoning into a shorter form, and one that is easier understood when dealing with "black or white" situations.
It was also confirmed for me by another course, which used much symbolism, but which I really had great trouble understanding. In the end, I had to treat the subject as if the basic foundations of it were being explained in terms of English, and not maths. Then it suddenly got really, really easy.
These sort of experiences and observations led me to realise that in almost every case that I came across, mathematical theorems and proofs could be explained in terms of English, but would take a lot longer, and that if we tried to explain mathematical proofs of theorems in ways that did not involve English at all, we'd also find it incredibly hard. So I realised that mathematics is taught in English, but using mathematical symbols just as a sort of shorthand, to refer to a pre-existing concept that is not found in common usage, with a single letter, to make the proof a lot more concise than it would be in English.
I don't need a formal proof for every theorem to prove it, because I don't need to see every case to be convinced of something. I don't need to examine every car, to know that if a car hits me at 50mph, it's likely to kill me, or hurt me really badly. I construct ideas based on my experiences, which includes things that happen to me, and things I study. So far, my experiences have shown me that every mathematical theorem seems to be able to be explained in English, and when it can't be, there is almost always a problem, either in lack of understanding of the theorem, or a flaw in the proof of the the theorem, and a genuine question if the theorem is true at all. | |
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| Infinity, does it extent both inwards and outwards ?
Posted: 11/1/2009 11:12:58 AM |
Constructionism doesn't always work, as sometimes I find that the particular constructions I was taught are so full of contradictions and counter-examples, that I have to abandon them
I quit reading at this point because you made it clear you don't understand the constructivist (intuitionist) viewpoint of mathematics. I suggest you do some reading on the matter, and how it relates to the concept of "infinity" as a "complete" mathematical object (to keep things consistent with the thread) | |
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| Infinity, does it extent both inwards and outwards ?
Posted: 11/1/2009 8:59:24 PM | Here are some good quotes I think are interesting...
But on the other hand to suppose that the infinite does not exist in any way leads obviously to many impossible consequences: there will be a beginning and end of time, a magnitude will not be divisible into magnitudes, number will not be infinite. If, then, in view of the above considerations, neither alternative seems possible, an arbiter must be called in.
—Aristotle, Metaphysics, Book 3, Chapter 6
Finally, let us return to our original topic, and let us draw the conclusion from all our reflections on the infinite. The overall result is then: The infinite is nowhere realized. Neither is it present in nature nor is it admissible as a foundation of our rational thinking - a remarkable harmony between being and thinking-David Hilbert...
Indeed, I think that there is a real need, in formalism and elsewhere, to link our understanding of mathematics with our understanding of the physical world. (A. Robinson)
There are at least two different ways of looking at the numbers: as a completed infinity and as an incomplete infinity. (E. Nelson [7])
During the renaissance, particularly with Bruno, actual infinity transfers from God to the world. The finite world models of contemporary science clearly show how this power of the idea of actual infinity has ceased with classical (modern) physics. Under this aspect, the inclusion of actual infinity into mathematics, which explicitly started with G. Cantor only towards the end of the last century, seems displeasing. Within the intellectual overall picture of our century ... actual infinity brings about an impression of anachronism. (P. Lorenzen[9])
(what is infinite about endlessness is only the endlessness itself)-Wittgenstein
(Thinking about all this infinity jargon is making me infinitely insane) | |
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| Infinity, does it extent both inwards and outwards ?
Posted: 11/1/2009 11:05:26 PM | | Infinity is just a word that describes a way of defining what one cannot and will not know. Infinity is actually a quite right and proper word to describe what is inherently a dance you will not attend. | |
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| Infinity, does it extent both inwards and outwards ?
Posted: 11/2/2009 12:02:14 AM |
It's speculation that infinity exists as well. It's speculation to say what happened in the past, or the future. It's speculation of what happens outside our solar system. But, science is founded through rigorous analysis of such speculations, and that is exactly what multiple infinities are all about, rigorous analysis.
no, you are confused here scorp. Saying what happened in the past and future is not blind speculation...we can determine with a high degree of accuracy what happened in the past (on the cosmic scale) by viewing what is going on now. In fact, by looking through a telescope, you are actually SEEING the past. We can't know for sure all the details about what has happened in the past, but we can get a strong sense of what happened grossly. This is far different from the kind of speculation that is required for making statements regarding "outside" our universe. It, by definition can not be a science at all, as there is no way to set up experiments to confirm theories. Therefore, it is, at best, philosophy, at worst, religion. By equating it with science, you in fact are insulting science. | |
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