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RECENT POSTS IN THIS TOPIC

Louis Kauffman: on 4/26/15 at 16:07pm UTC, wrote Dear Bill, Thank you for the kind comments. Holocosm is our term for that...

William Parsons: on 4/17/15 at 18:24pm UTC, wrote Hi Lou and Rukhsan-- I loved your essay. Well written and so handles a...

Armin Nikkhah Shirazi: on 4/14/15 at 22:20pm UTC, wrote Dear Louis, Your enthusiasm and passion for the subject matter is really...

Louis Kauffman: on 4/8/15 at 16:25pm UTC, wrote Dear Joe Fisher, I will certainly read and respond to your essay! I think...

Joe Fisher: on 4/8/15 at 15:38pm UTC, wrote Dear Louis, I think Newton was wrong about abstract gravity; Einstein was...

Louis Kauffman: on 3/16/15 at 20:05pm UTC, wrote Dear Chris, Of course I am modeling. I do not regard this as an...

Christopher Duston: on 3/16/15 at 14:02pm UTC, wrote Dear Louis and Rukhsan, Thanks for a great entry - I think I...

Robert MacDuff: on 3/16/15 at 1:47am UTC, wrote Lou The point that I was attempting to make is that mathematics, since it...


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FQXi FORUM
October 17, 2019

CATEGORY: Trick or Truth Essay Contest (2015) [back]
TOPIC: Trick or Truth: The Mysterious Connection between Mathematics and Physics by Louis Hirsch Kauffman and Rukhsan-Ul-Haq Wani [refresh]
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Author Louis Hirsch Kauffman wrote on Mar. 6, 2015 @ 16:45 GMT
Essay Abstract

The essay is in the form of a dialogue between the two authors. We take John Wheeler's idea of "It from Bit" as an essential clue and we rework the stucture of the bit not to the qubit, but to a logical particle that is its own anti-particle, a logical Marjorana particle. This is our key example of the amphibian nature of mathematics and the external world. We emphasize that mathematics is a combination of the conceptual and the calculational. At the conceptual level, mathematics is structured to be independent of time and multiplicity. Mathematics in this way occurs before number and counting, and can be described by the world of logic and boolean arithmetic. From this timeless domain, mathematics and mathematicians can explore worlds of multiplicity and infinity beyond the apparent limitations of the physical world and see that among these possible worlds there are matches with what is observed.

Author Bio

Louis Hirsch Kauffman is a Professor of Mathematics at the University of Illinois at Chicago. He is a topologist working on knot theory and the structure of form. He is the author of books on knot theory and physics and is the originator of state summation models for knot invariants that relate these invariants to partition functions in statistical mechanics. Rukhsan-Ul-Haq is presently a phd student working in the field of strongly correlated electron systems in JNCASR Bangalore India. He is very fascinated by the relation between physics and mathematics. He is also interested in the foundations of physics and mathematics.

Download Essay PDF File

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Cristinel Stoica wrote on Mar. 6, 2015 @ 21:13 GMT
Dear Lou and Rukhsan,

I like the idea pointed out by Spencer-Brown, that "the Universe is constructed to be able to see itself, and we are constructed to be able to see the Universe". This is probably because otherwise we couldn't adapt and survive. It is interesting the notion of "distinction", and its expression in terms of sets. Also, you emphasize the role of numbers, but also of...

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rukhsan ul haq wani replied on Mar. 10, 2015 @ 17:05 GMT
Thanks Dear Cristi Stoica for your comments.

What is quite amazing is that the urge in a physicist/mathematician to understand the universe is not his own(because he is just a part of the whole) rather it comes from the universe herself.Universe seeks to understand herself.It is this urge which is the base of the quest in physics and mathematics.

Now the curious thing is that for universe to understand herself she needs an observer and that is where "distinction" comes.As observer makes distinction and universe get created and with that "universe of discourse" also takes birth.The formal properties of distinctions are same as that of projection operators in quantum mechanics which are operators for making measurement.Mathematics follows from these distinctions.Both Boolean logic and arithmetic can be derived from the laws of distinctions/laws of form.But what is mots amazing is that algebra of Majorana fermions and fusion rules of Fibonacci anyons also arise from the same ground.So symbolically what happens in a distinction is that a mirror get created between observer and the observed on which unfolds the recursive play of forms and out of this created duality time is born.The eternal thing is that there is unity and duality is just emergent.Observer and observed are tied in a knot.

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Author Louis Hirsch Kauffman replied on Mar. 12, 2015 @ 22:31 GMT
My point of view is that observer and distinction arise together. You can ask how this comes about,

but at the bottom this is ones experience. Halls of mirrors and complexity arise naturally enough once there appear to be distinctions. But distinctions are, like coordinates, telling a tale from a particular point of view. You have to start with particular points of view and work hard to find out what is common to them. Spencer-Brown says "We take as given the idea of distinction and the idea of indication and that one cannot make an indication without drawing a distinction. We take therefore the form of distinction for the form." This makes "the form" into a circularity or a quest for "the form of distinction"! It is a cybernetic point of view to show a meaning through a circularity. I particularly like Heinz von Foerster's sentence "I am the observed link between my self and observing myself.".

All such circularities are invitations to enter into the spiral experience of working through the circle and returning at a different place. Differential geometry measures curvature by just that, and here we are looking at the holonomy of the cybernetics of cybernetics.

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Michel Planat wrote on Mar. 7, 2015 @ 14:26 GMT
Dear Lou,

It is always a real pleasure reading your well written and ambitious papers. In your essay with Rukhsan, you succeed in relating, in simple words and concepts, the Majorana particle concept to the permeable boundary between physics and maths "the very foundations of physics, mathematics and the roots of thought.".

I use a different approach motivated by my continuous interest for the Kochen-Specker theorem. Since the braid group B3 modulo its center is also the modular group, there are clearly connections to my last paragraph. I also know that the general braid group is connected to dessins d'enfants in a highly non trivial way.

Best wishes

Michel

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Author Louis Hirsch Kauffman replied on Mar. 12, 2015 @ 22:39 GMT
Dear Michel,

Thank you! I shall look at your papers. I am particularly fascinated by how a representation of the

circular Artin braid group arises from Majorana operators. Suppose you have $c_1, ..., c_n$ with

$c_{i}^{2} = 1$ and all different pairs anti-commute. Let $\sigma_{i} = (1 + c_{i}c_{i+1})/\sqrt{2}.$

Then the $\sigma_{i}$ satisfy the braiding relations. This is a fundamental representation of the braid group that is just underneath the usual representation for Fermions. Perhaps you have your own point of view on this.

Best,

Lou K.

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Author Louis Hirsch Kauffman replied on Mar. 12, 2015 @ 22:42 GMT
I meant

$\sigma_{i} = (1 + c_{i+1}c_{i})/\sqrt{2}.$ with $\sigma_{n} = (1 + c_{1}c_{n})/\sqrt{2}$ to make it

wrap around and become a representation of the circular braid group.

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Robert MacDuff wrote on Mar. 8, 2015 @ 04:56 GMT
Hi Lou, it has been a few years.

Glad to see you are still expanding on G-Spencer Brown’s Laws of Form. Such a curious thing, creating something out of nothing.

However I would like to ask a question and I can think of no one better to provide an answer. I am not sure this is the forum to do so but I thought I might try. My question is about your example of an array of dots. Is this not a classic example of something that mathematics can’t model?

You claim the answer is 289d (d=dots). You can’t be assuming that 17d horizontally times 17d vertically because that would mean, mathematically that the answer should be 289 d2. Another reason is that it seems that 17d + 17d should be 33d unless you want to add one twice.

Or are you making the logical argument that 289 dots has a relationship to a group of 17 dots and the group of 17 dots has a relationship to 1 dot thus logically 289 dots should have a relationship to 1 dot, which symbolically could be encoded as follows: {(289d/17d) x (17d/1d) = (289d/1d)}. This matches up very nicely with your 172 = 289. However it leaves the curious question as to what 17 + 17 might be, as one encodes the Whole/group relationship and the other encodes the group/dot relationship.

cheers

Rob MacDuff

Lou, i do have an answer but I would like to see your comments.

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Author Louis Hirsch Kauffman replied on Mar. 12, 2015 @ 22:48 GMT
Dear Rob,

We usually interpret 3 x 3 as 3 rows of three dots. That makes 9 dots altogether.

***

***

***

What are you thinking here? It would seem to be something else!

Why is 2 x 3 = 3 x 2? A good answer in this model is to rotate the rectangle by ninety degrees.

Concrete arithmetic is a good place to examine how distinctions that have some memory associated with them generate arithmetic. When we start working more abstractly and want to talk about

numbers like 2^{2^{2^{2^{2^{2^{2}}}}}}}, then there is no way to keep using dots unless you have an ideal notion of dots. So the principle of mathematical induction takes over.

Best,

Lou

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Robert MacDuff replied on Mar. 16, 2015 @ 01:47 GMT
Lou

The point that I was attempting to make is that mathematics, since it is mainly based upon ordered sets of symbols, is limited in the types of structures it can naturally describe without being extended by adding units or alternate types of numbers. It seems to me, that mathematics uses two different types of numbers indiscriminately: cardinal and relationship. To easily see this requires extending mathematics by including units. 3 in three rows is a cardinal number and 3 in the number of dots per row is a relationship number.

3r x 3d/1r = 3r x (3/1)(d/r) = 3r x (9/3)(d/r) = 9d.

The standard proof that 3 x 3 = 9 assumes 3 x 3 = 3 + 3 + 3. However using both cardinal and relationship numbers 3 x (3/1) = 3 x (9/3) = 3/3 x 9 = 1 x 9 = 9. So we can interpret 3 x (3/1) as 3 rows where there are three dots per row and thus 9 dots in total. I would say that this models the array in a more precise way.

Is the distinction between cardinal and relationship numbers useful? It matches very nicely with the way in which science uses operations, especially as multiplication can’t be interpreted as multiple addition but many can as a relationship times a quantity. Relationship numbers enable the encoding of multiplicities but they also act as recursive functions that map numbers back onto themselves. In other words they encode how numbers are related to one another. It also puts multiplication and division on their own foundation rather than being tied to addition. If multiplication did not introduce a tacit alternate type of number then 3 + 3/1 would make sense within the context of the problem.

Lou, it seems as if this would allow multiplication in your set of numbers. 3 X (3/1) to get 9, where 3 = {{ } + {{ }} + {{{ }}}} 3/1 = {{ } + {{ }} + {{{ }}}} /(( }}.

Cheers

Rob

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Joe Fisher wrote on Mar. 12, 2015 @ 20:59 GMT
Accurate writing has enabled me to perfect a valid description of untangled unified reality: Proof exists that every real astronomer looking through a real telescope has failed to notice that each of the real galaxies he has observed is unique as to its structure and its perceived distance from all other real galaxies. Each real star is unique as to its structure and its perceived distance apart from all other real stars. Every real scientist who has peered at real snowflakes through a real microscope has concluded that each real snowflake is unique as to its structure. Real structure is unique, once. Unique, once does not consist of abstract amounts of abstract quanta. Based on one’s normal observation, one must conclude that all of the stars, all of the planets, all of the asteroids, all of the comets, all of the meteors, all of the specks of astral dust and all real objects have only one real thing in common. Each real object has a real material surface that seems to be attached to a material sub-surface. All surfaces, no matter the apparent degree of separation, must travel at the same constant speed. No matter in which direction one looks, one will only ever see a plethora of real surfaces and those surfaces must all be traveling at the same constant speed or else it would be physically impossible for one to observe them instantly and simultaneously. Real surfaces are easy to spot because they are well lighted. Real light does not travel far from its source as can be confirmed by looking at the real stars, or a real lightning bolt. Reflected light needs to adhere to a surface in order for it to be observed, which means that real light cannot have a surface of its own. Real light must be the only stationary substance in the real Universe. The stars remain in place due to astral radiation. The planets orbit because of atmospheric accumulation. There is no space.

Warm regards,

Joe Fisher

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Joe Fisher replied on Mar. 12, 2015 @ 21:12 GMT
oops, I forgot to paste the first part of the letter.

Dear Professor Kauffman,

You wrote: “Thus you hear mathematicians working with higher orders of infinity, and even hoping to approach the meaning of absolute infinity. All this

is possible by working from the holocosm where there are no numbers, no multiplicities, no infinities and no finity other than zero and one, nothing and something.”

All that needs to be done then is to remove the zero from mathematics and the abstract “nothing” from physics and good sense will emerge.

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Author Louis Hirsch Kauffman replied on Mar. 12, 2015 @ 22:54 GMT
Dear Joe,

I think that nothing will come from attempting to remove nothing. Nothing is our most practical concept. It stands for those clearings that we find or create that allow us to work, construct and perceive. I really appreciate that sense of uniqueness that you so beautifully express about every perception and every perceived phenomenon. I fell that way when I "look" at the empty set.

So perfect and unique it is. And there can be only one empty set. For if two sets were empty, then they would have exactly the same members, namely none! And two sets are equal exactly when they have the same members.

Best,

Lou K.

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Lorraine Ford wrote on Mar. 12, 2015 @ 23:04 GMT
Dear Louis and Rukhsan,

I think your essay is very relevant, which is not to say that I entirely agree with everything you say.

I think that physical reality has an "inner dynamic" quality, and that aspects of the dynamic RELATIONSHIPS of physical reality could be symbolized by a Not operator. But I can't quite see that this symbol "can be seen as a "logical particle" whose counterpart in the mathematical physical world is a Majorana Particle".

I like your idea of a "distinction":

"We begin, not with mathematics as a known formalism, or with physics as laws expressed in mathematical form, but with the condition of an observed world, a world in which it is possible to have a division of states into that which sees and that which is seen. One can begin with the idea of a distinction…"

And I also liked the following passages:

"A host of ideas and mathematical ways of geometrizing are combined to make the concept of the electron useful and matching with the actions and observations of experimentalists. Simple localized objects have disappeared...I can give you an example that is closer to home…for the mathematician [a] knot exists in the eternal holocosm of non-numerical forms. There is a desire to make this holocosm the basis of the physical world. I cannot assent to that unless we explore how ideal entities like numbers and knots are related to our experiences. "

"In this sense, mathematical concepts are the basis of our experience. "

"The physicist is inseparable from the Universe herself. It is the Universe that studies herself through the articulations of mathematics and the observation of experience..."

I have a quote from Louis' article "What is a Number?" in my essay "Reality is MORE than what maths can represent" – using it in connection with one of my arguments: that numbers represent fundamental physical structures, but sets don't.

Cheers,

Lorraine Ford

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Author Louis Hirsch Kauffman replied on Mar. 13, 2015 @ 03:23 GMT
Dear Lorraine Ford,

Let me explain what I mean by saying that the mark is a logical particle. For typographical purposes lets use < > for the mark. Then its formal properties are < > < > = < > and = . Here the = sign means "can be replaced by" and the blank space is a blank space. We could use # to stand for a blank space. Then # would have the formal properties...

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Author Louis Hirsch Kauffman replied on Mar. 13, 2015 @ 03:25 GMT
Right at the beginning a typo!

< > < > = < >

=

And we can take = #

where # is a symbol to stand for the absence of any symbol.

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Author Louis Hirsch Kauffman replied on Mar. 13, 2015 @ 03:29 GMT
The typo was more serious than I thought.

In this word processor a mark inside a mark automatically vanishes, or so it would seem.

I want to let M = < > and write < M > = #. But when I write it directly it vanishes. Lets try and experiment. Use { } for the marked state. Then the interactions would be

{ } { } = { }

{{ }} =

or

{{ }} = #.

All this illustrates how issues of language and symbols always interact with attempts to express mathmatics.

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Author Louis Hirsch Kauffman wrote on Mar. 15, 2015 @ 15:07 GMT
This is a reply to Lorraine Ford's post of March 14, 2015.

1. The mark as a sign such as { } makes a distinction in its own form. If it refers to a physical state or

to a mathematical state, then it may refer to something quite different from itself. For example the sign for two, 2, refers to pairs, but 2 as a sign is connected and not a pair! If I decide that the numbers are represented by |, ||, |||, ... then each sign has the property of the number to which it refers. This iconic nature of some signs is important, but we cannot insist upon it. Thus the sort of

distinction that a sign makes is important but may have a different importance from its referent.

2. How can something arise from nothing? All stories we tell turn the 'nothing' into a something from which other somethings can arise. For example. Consider a smooth flat woven price of cloth. This can be the 'nothing'. Then folding the cloth or crumpling it gives rise to many remarkable forms. These forms arise from the nothing of the cloth. Nothing means no thing and that means no object relative to some way of discerning or creating objects.

3. I prefer to take the initial state (if there is one) as not distinguished and the states that arise 'from it' as the result of processes of discrimination. This is the mathematical point of view where we start from very little and construct mathematical universes by making definitions. Nobody says that Nature does this except when she is being mathematical!

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Christopher Duston wrote on Mar. 16, 2015 @ 14:02 GMT
Dear Louis and Rukhsan,



Thanks for a great entry - I think I understand the core of what you are saying but I'm not sure you are doing anything more fundamental than "modeling reality".

For instance, you begin with the tenant that "the world is constructed in such a way that we can see itself". However, later in the essay you make a point about clarity of language - working out "what every word in that question means" (page 5) and the word "understood" (page 6). It seems to me these are at odds with each other - if the world can be constructed to understand itself from basic logical system (such as the Calculus of Indications), why do you even have to mention the understanding of these words? Shouldn't our own understanding of them be universal?

As for the Calculus of Indications, you imply that it's just an extension of the Boolean algebra, but isn't it actually a *drastic* modification (even destruction) of this? By making the operator also a state (first equation on page 8) you are really constructing a fundamentally different version of reality. It would appear you shift from an axiom-based reality to a reality where the differentiation between axioms and theorems is meaningless. But isn't that just another hypothesis for a model of reality?

To summarize my point of view, I think you are modeling a reality using "the mark" as "a bit", but isn't it still just a model? We can have the same universal understanding of particles constructed in this way as if we modeled electrons using a quantum field with specific properties. So I'm not sure you're doing anything different than "modeling reality".

Anyway, it was a great, thought-provoking read. I wish you both luck in the contest!

Chris Duston

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Author Louis Hirsch Kauffman replied on Mar. 16, 2015 @ 20:05 GMT
Dear Chris,

Of course I am modeling. I do not regard this as an epistemological error. In fact I regard the unrestricted identification of physical reality with mathematics as a serious epistemological error.

This does not mean that there are no places where the mathematics and the physical are indistinguishable. When they are indistinguishable, it means that there is an awareness...

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Joe Fisher wrote on Apr. 8, 2015 @ 15:38 GMT
Dear Louis,

I think Newton was wrong about abstract gravity; Einstein was wrong about abstract space/time, and Hawking was wrong about the explosive capability of NOTHING.

All I ask is that you give my essay WHY THE REAL UNIVERSE IS NOT MATHEMATICAL a fair reading and that you allow me to answer any objections you may leave in my comment box about it.

Joe Fisher

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Author Louis Hirsch Kauffman replied on Apr. 8, 2015 @ 16:25 GMT
Dear Joe Fisher,

I will certainly read and respond to your essay! I think however that your post is not quite appropriate, since one is supposed to comment on the essay that we wrote in this space. However, first without referring to your essay, I want to examine from our point of view what it would mean to say that the real universe is not mathematical. From my point of view all linguistic constructions accompanied by concepts are mathematical. David Finkelstein once said "Mathematics is a form of literary endeavor

with a very special type of criticism.". When one goes to the basis of mathematics and finds that it is about the properties of distinctions then we see that mathematics always involves awareness/mind/concept. It is not bare formalism or mechanical calculation. Given that stance, it is clear that I cannot say that a world without concepts is a mathematical world. If I believed in a purely material world in this sense, then I would have to say that 'it' knows no mathematics except through the observers of that world and while the universe may apparently follow mathematical rules, that is our description of it and not it itself. But if you take the point of view that there is in the Reality an intertwining of concept/awareness and materiality, then the universe herself could be a great mathematican. Even in this view, the mathematics has to to do with the dialogue of the Universe with herself.

Now I will go read your essay and make comments on it on your site!

Best

Lou K.

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Armin Nikkhah Shirazi wrote on Apr. 14, 2015 @ 22:20 GMT
Dear Louis,

Your enthusiasm and passion for the subject matter is really shining through in this dialogue. I had thought of the division of mathematics into the conceptual and the calculational as an artifact of how we humans conceptualize what mathematics is about, not as something that is intrinsic to the subject, and the main reason for that was that it seems difficult, if not impossible, to draw exact boundaries between the two. Your dialogue exposed me to an alternative viewpoint that I'll still have to think some more about.

To give some critical feedback, I would have appreciated a little more in the way of explanation of the Calculus of indications. In particular, I found myself wondering whether the difference in the thickness of the lines of the sideways L in different equations signified anything, and also wondered about the meaning of the dot in some of the equations, since negation had only been defined in terms of the combination of the L and the dot.

As a topologist, you are very close to the foundations of mathematics and therefore might possibly be interested in finding out a little about my current effort to extend the foundations in order to increase the expressive power of mathematics, some of which is outlined in my entry. My background is actually in physics but I found that certain ideas and concepts I entertain pertaining to quantum mechanics do not seem to be formally expressible using the language of contemporary mathematics. I would certainly appreciate any critical feedback from an expert mathematician. Who knows, perhaps you might even find this to be an exciting venture into uncharted corners of the holocosm.

Best wishes,

Armin

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William T. Parsons wrote on Apr. 17, 2015 @ 18:24 GMT
Hi Lou and Rukhsan--

I loved your essay. Well written and so handles a complex topic well. As a general rule, I don't like dialogues (after all the poor knock-offs written in response to D. Hofstadter's GEB), but you guys managed to pull it off. More importantly, you offered the kind of "outside the box" approach that I think this contest intended to inspire. Your essay was very, very thought-provoking. While I don't agree with everything you wrote, you certainly got my attention.

Quick question: Could you describe a bit more what you mean by the term "holocosm"? Is this the realm of pure logic or a Boolean realm? What is its ontological status, in your view?

I am a bit puzzled by the relatively low rating of your essay by the community. I shall add my vote and seek to rectify.

Best regards,

Bill.

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Author Louis Hirsch Kauffman replied on Apr. 26, 2015 @ 16:07 GMT
Dear Bill,

Thank you for the kind comments. Holocosm is our term for that eternal timeless world,highly creative, that mathematicians imagine as existent. Concepts and ideas are real and living in the holocosm. This can only be metaphorical in relation to common notions of existence and we see the contrast when we examine how we talk about mathematical constructs. Many people imagine that...

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