Infinities in literature and mathematics

_some_infinities_are_bigger_than_other_infinities__by_aguynamedlewis-d6byxiyby Jorge Alejandro Laris Pardo

During this past month, I was having a conversation with a couple of friends who study Latin-American Literature, and I noticed that they were having a hard time understanding how a literary work can have infinite critical interpretations, while at the same time not all its interpretations are critical. Apparently they found this to be contradictory.

I was shocked by their confusion, because to me the idea in question is almost self-evident. But later I came to acknowledge the fact that my friends, who are schooled in the humanities, have little if any notion of the mathematical idea of the infinite. For that reason, I suggest in this essay that the humanities can learn something from the concept of infinities in mathematics.

The problem with Romanticism’s concept of the Infinite

According to Alain Badiou, the history of Western philosophy can be divided into two great periods. First, the era before and including Kant, when mathematical reasoning was considered a singular way of thinking that interrupted the predominance of opinion — or, to put it in philosophical jargon, of Doxa — in philosophical reasoning. And second, the post-Kant era, which gave birth to Romanticism, which was consummated by Hegel, whose philosophical system is powered at its core by the schism between math and philosophy. Following Badiou [1], this schism also lies at the core of 19th century positivism and modern radical empiricism — because arguments put forth by these movements just flip to the other side of the same coin without really solving the problem — and has greatly impacted contemporary thinking, especially in the humanities.

In fact, for Hegel, nature is death Logos, or ideas, and as such it has no real value for human understanding. For him, math was part of nature, and therefore it too has no real meaning for human understanding of the world. He even went as far as to criticize the use of math in physics. For example, he once wrote:

“Whoever approaches this part of physics [Newtonian physics] soon realizes that it is rather a mechanics than a physics of the heavens and that astronomy’s laws derive their origin from another science, from mathematics, rather than actually having been teased from nature or constructed by reason […] All physicists before him [Newton] regarded the relationship between the planets and the sun as a true one, i.e. as a real and physical force. What Newton did was to compare the magnitude of gravity shown by experience for bodies forming part of our earth with the magnitude of celestial motions; he then proceeded to deal with everything else using mathematical reasoning from geometry and calculus. We must be especially wary of this binding of physics with mathematics; we must beware of confusing pure mathematical grounds with physical ones; namely, of blindly taking lines deployed by geometry as helps to construction in proving its theorems for forces or force directions.” [2]

He goes on to describe gravity in terms of planetary needs and desires, as if planets  themselves were capable of some kind of reasoning. Having said that, Hegel declares that the real understanding of the infinity cannot come from math, but has to originate from philosophy (as he understood it).

Following this principle of separating math from philosophy, Hegel develops an idea about infinity that identifies it with the Unity and the Absolute. This idea is widely spread today — it is held by my literary friends, to say, among others — and it has closed the possibility for an entire discipline to grasp the idea of multiple infinities.

For Badiou, the true reason lying behind this schism between math and contemporary humanities is to be found in the idea of historicism. According to historicism, the content of every idea is inescapably attached to the historical context in which it originated. Therefore, all “truths” are such only within a particular historical framework. While this idea may be true for many areas of inquiry, it is definitively not true for mathematical reasoning, precisely because the latter is a type of human reasoning that transcends the bindings of history. In other words, math has shown us that human thinking is not condemned to always be a matter of opinion.

Mathematical infinities

Thinking about infinity can be a source of really severe headaches. For example, let’s think about integers (…-4, -2, -1, 0, 1, 2, 3, 4… and so on). It is fascinating to note that there are as many integers as there are prime numbers (numbers that can only be divided by 1 and by themselves, as 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97…). Surely this is fascinating, but it is also puzzling! I mean, between 1 and 100 we can find 100 integers and only 25 primes. Then, how is it possible to argue that infinity contains as many integers as it contains primes?

There is an easy way to prove this, and it was proposed by 19th century mathematician Georg Cantor. He first stated that it is possible to count two different sets of things by matching a one to one correspondence between their elements. For example, let’s imagine that I ignore how many fingers I have in my right hand, as well as how many platonic solids there are. Let’s suppose, too, that I never attended school and I don’t how to count. Nevertheless, I’m a stubborn person and still find a way to know how many platonic solids there are in relation to the fingers in my right hand, by the simple means of matching a one on one correspondence between them. One of the possible outcomes of this exercise will be as follows:

fig1

By doing so, now I know that there are at least as many platonic solids as fingers in my right hand.

What Cantor discovered is that it is possible to measure infinities in exactly the same way; as shown in the picture below:

fig2

Thanks to this method, Cantor was able to determine that in fact, there are as many integers, as there are prime numbers. Cantor named these kind of infinities countable infinities — although, as Jame Grime has pointed out, they would be better named enlisting infinities [3]and proved that every countable infinity is as big as the infinity of the integers.

In the next picture, it is shown how it is possible to enlist fractional numbers, too. Therefore, it is possible to assert that there are as many fractional numbers as primes and integers:

fig3

This observation in itself represented a big breakthrough, but what Cantor also found is that there are some types of infinities that simply cannot be enlisted. He called them uncountable infinities. The real numbers are an example of these kind of infinities. To be able to understand this principle, I unsuccessfully tried to enlist all the numbers between 0 and 1. As you can see in the picture:

fig4

By doing a simple exercise, Cantor also proved that in any given list of real numbers, we could always find a number that is not contained in it; and that we can do this by following three simple rules: 1) In writing the first digit of our number we will pay attention to the first digit of the first number of the list; then, in writing the second one, our attention should be on the second digit of the second number of the list… and so on; 2) if the number we are looking at is not a 1, we are going to write down a 1; 3) but if the number we are looking at is a 1, then we are going to write down a 2. The result of this procedure will always be a new number. Don’t believe me? Try it!

fig5

After that, Cantor showed that the uncountable infinities are in fact bigger than the countable ones! So, there are some infinities that are bigger than others! This clearly goes against Hegel’s idea of a Unique and Absolute Infinity.

But hold your breath. The most exciting part of the story is yet to be told, because Cantor himself was convinced that there should be something between the countable infinities and the uncountable ones. This idea he called the hypothesis of the continuum, and he could never prove it. In 1938 Kurt Gödel proved that it is actually impossible to show that this hypothesis is false; and, even more amazingly, in 1963 Paul Cohen proved that it is also impossible to prove it correct! So we are condemned to live with this ignorance forever!

Now that we have learned some basics of what mathematicians think when they think about the infinity we can finally discuss how this idea can affect the philosophy of literature.

Infinite interpretations of literary works

One of the main things critics of literature do is to interpret literary works. In the past, the notion that a literary work, say a novel, only has one real meaning was widely accepted. But, with the dawn of positivism and the ascension of hermeneutics in the humanities, that notion was left behind. Nowadays almost every critic of literature will argue that a novel has as many meanings as there are interpreters. Most of them will even argue that a novel has, in fact, potentially infinite meanings.

But, if every reader can find different valid meanings in a novel, then, how is it that the work of the literary critic is still relevant? According to French philosopher Paul Recoeur [4], most of the interpretations given to a novel by ordinary people are just quick guesses based in conjectures. For sure, they are valid, but clearly are not as valuable as that of the literary critic, who spends most of her time studying the history of literature and its relation with the social contexts in which it is produced and received.

Also, a critic has to present her interpretation to a whole group of other critics. In doing so, she has to defend her ideas about a novel or a poem with a number of arguments: by relating it to the culture where it was produced, with biographical information about the author, by paying attention to what other people have believed about that specific novel or poem in different moments of history, or even by studying the reasons the publisher had to publish it in the first place.

For all of that, the critic’s interpretation of a literary work tend to be better informed and argued than the ones of casual readers. As Recoeur said, maybe there are no methods to make valid interpretations, but there surely are ways to make those interpretations invalid.

To help clarify this idea, I will call interpretations of casual readers real interpretations, in analogy to the uncountable infinity of the Real numbers; and I will refer to the interpretations given by literary critics as integer interpretations, in clear analogy to the countable infinity of Integer numbers. Is there any connection between the two types of interpretations? How could we even know!

The main difference between real interpretations and integer interpretations of a literary work is that the first ones are more numerous, easier to produce and, hence, less valuable than the second ones. This is so, because more hard work is needed to produce an integer interpretation than it is to produce a real one (the only ingredient needed for the latter is a mere guess). Georg Cantor paraphrased: the integer numbers are like the bright stars in the sky at night, while real numbers are darkness.

Of course, this is an analogy. Philosophy and math are based on different kinds of reasoning, and I’m pretty confident they will stay this way forever. My general point is that some times, analogies can help us mediate between different kinds of knowledge. And especially, that we should be aware of not confusing infinity with the absolute: in the humanities there may be no “true” interpretations, but clearly some interpretations are better than others.

A final thought about the relation between math and literature

Why should we talk about math in an essay about literature? I have two distinct reasons for this. In the first place, because, in the end, the value of literature and mathematics is that they both enrich our life. Life itself is meaningless but it acquires value to us because we, somehow, have created ways to enjoy our individual and collective existences. And I can’t see any reason to doubt that art, scientific knowledge and thinking itself are noble ways to enrich one’s life.

In contemporary culture, many people value things, including scientific knowledge, according to how practical they are. What we don’t always realize is that while we are wishing for a scientific notion to be useful in a tangible way, what we really want is for it to be relevant to our lives in a way that may contribute to enriching our existence. The most important thing about the Theory of General Relativity is not that it enables us to use GPS systems (which are pretty useful nonetheless, if you ask me); rather, it is important because it is fascinating to grasp it. To understand that, as far as we know, space can be curved and the flow of time can be altered by the effect of mass.

The same thing applies to math. As the mathematician Harold Harley famously stated, “very little of mathematics is useful, and this minority is comparatively dull” [5]. Something like that can be said about art and, in this sense, math and literature can be measured with the same ruler.

Now, I’m not going as far as Alain Badiou when he states that math is ontology itself [6], but I can argue that mathematical thinking can be of great value to metaphysics. This is because, as we have seen, math transcends the limits imposed by doxa and by historicism. Math can also also take the form of a secular truth, at the least according to some recent speculations about the meaning of “nothing” [7].

But, as Badiou — and Plato well before him — has stated, even when math offers true knowledge or episteme, it alone cannot suffice for wisdom. For mathematical knowledge to turn into wisdom it is necessary to include philosophy, and literature.

_____

Jorge Alejandro Laris Pardo is an undergraduate student in the History of México at the University of Yucatán. He is currently working on a thesis about “Science and the Polemics around the Idea of Science in the Yucatan Peninsula in the Times of the Restoration of the Republic (1867-1882),” were he is concerned about studying how social contexts can affect science and the quest for scientific knowledge, without falling into the extremes of postmodernism.

[1] I highly recommend Conditions, by A. Badiou, 2009.

[2] Hegel, De Orbitis Planetarum.

[3] Infinity is bigger than you think.

[4] Interpretation Theory: Discourse and the Surplus of Meaning, by P. Ricoeur, 1976.

[5] A Mathematician’s Apology, by H. Hardy, 2005.

[6] Although some mathematicians do!

[7] The Book of Nothing: Vacuums, Voids, and the Latest Ideas about the Origins of the Universe, by J. Barrow, 2002.

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49 replies

  1. “…Cantor himself was convinced that there should be something between the countable infinities and the uncountable ones. This idea he called the hypothesis of the continuum, and he could never prove it. In 1938 Kurt Gödel proved that it is actually impossible to show that this hypothesis is false; and, even more amazingly, in 1963 Paul Cohen proved that it is also impossible to prove it correct! ..”

    To be accurate, this must be reversed. So one of the following needs to be done:

    exchange “correct” and “incorrect”; or
    exchange “Godel…1938” and “Cohen…1963”; or
    change “something” to “nothing”.

    Perhaps the essay could also have mentioned that Cantor proved more generally that no infinite cardinal is largest, with a beautiful proof that is, in a subtle sense, the same idea as that the author here gives for countable versus continuum.

    And “Harly” in the text (not refs) should be “Hardy”, as in refs.
    But Hardy is considered quite wrong on that point by virtually everyone today. The example often given is the application of his very own subject, number theory, in encryption. I don’t know whether he was unaware of Riemann’s geometry being ‘used’ in General Relativity, or simply thought Einstein was wrong, to give another counterexample. Etc., ad infinitum, to jocularly refer to the essay’s main topic, and to end with a non-sentence!

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  2. Maybe better analogy would be reals vs rationals: rational numbers are dense subset of real numbers and have the same cardinality as integers; that makes real numbers separable set.
    That means every real number can be approximated arbitrary close with rational numbers, or (not quite in the spirit of article) every layman’s interpretation can be approximated arbitrary close with critics’ interpretations.

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  3. One could live in a world where one would never need infinity at all (using a particular typed[1] versus untyped language). Perhaps there are beings[2] that live in such a world. Joel David Hampkins says there is a multiverse[3] of mathematical truth in (infinitary) set theory. My own thought about infinity is about whether we can ultimately find or make hypercomputers[4]. It would have been interesting if they had used one in the movie “Interstellar”.

    [1] http://plus.google.com/+PhilipThrift/posts/GGeDVVTX1Mg
    [2] http://poesophicalbits.blogspot.com/2012/04/persons-without-infinities.html
    [3] http://jdh.hamkins.org/the-set-theoretic-multiverse/
    [4] http://en.wikipedia.org/wiki/Hypercomputation

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  4. From a mathematician’s point of view, the term “infinity” is used far too loosely in the humanities, and also often in science and philosophy. It is commonly used for things that are not actually infinite, but merely have upper bounds that can’t easily be identified. For example, it is often said that the English language allows for an infinite number of possible sentences — but I can confidently assert that nobody ever has or ever will generate a sentence with even a billion words; and in comparison to infinity, a billion words is infinitesimal. The same thing applies to critical interpretations. Finite beings are not capable of encompassing actual infinities. This distinction might come across as pedantic, but in computer science the difference between the infinite and the merely large has great practical importance.

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  5. Replying to Bill S.,

    “.. it is often said that the English language allows for an infinite number of possible sentences — but I can confidently assert that nobody ever has or ever will generate a sentence with even a billion words..”

    It is important to distinguish between infinitely many sentences and infinitely lengthy sentences. The latter is pretty much excluded, if not theoretically out of the question. But the former is clearly true of English, as in the example:

    I am. I am and I am. I am and I am and I am. …etc….

    This requires the existence of sentences of arbitrary, but finite, length. Maybe that was the point being made.

    But certainly not uncountably many sentences (he says in a non-sentence once again!)

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  6. Rather a long article to correct a basic misunderstanding of infinity, but some fun observations along the way besides so it’s hard to hold it against the author.

    > and it has closed the possibility for an entire discipline to grasp the idea of multiple infinities.

    Seems like a bit of an over-generalisation. The friends the author talked to may have misunderstood the concept of infinity, but I’m not sure the same is true of the entire discipline.

    Like Darko Mulej I have a problem with the analogy drawn, but in a different way. The basic misunderstanding your friends exhibited was thinking that an infinite set must include all elements. It is not necessary to get into the different cardinalities of infinity to illustrate that this is false, and as far as I can see there is no reason to think that critical interpreations have a different cardinality than casual interpretations.

    A much better analogy, in my view, would have been to two countable infinite sets, one of which is a superset of the other. For example, I would have chosen the set of rational numbers as analogous to the set of all valid interpretations, and the set of integers as analogous to the set of all valid critical interpretations. Both sets are infinite. One set is entirely contained in the other. The cardinalities of both sets are the same, and yet there are elements in one set that are not in the other. Cantor’s proof is of course worth knowing about, but it has nothing to do with the reasons the author’s friends were mistaken and is not needed to demonstrate the error.

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  7. “It is fascinating to note that there are as many integers as there are prime numbers” presumably because there is an infinite number of both; infinity = infinity, therefore the numbers are equal.

    As an idiot (from the Greek ‘idiotes’, a layperson or individual) I would point out that the concept of infinity EXISTS in the minds of many imaginative people, but the concept does not correspond to anything REAL. The universe is real and apparently it is finite. The universe is vast, beyond our comprehension, but there is not anything in it that is infinite. Mathematicians operate in an abstract universe, an alternate universe. It is a universe that only exists for a particular mathematician, but that doesn’t make it so. (Despite the fact that some opinions are supposedly more valuable than others.)

    Abstract thoughts about zero or infinity, like many other abstract thoughts, exist in our minds; the thought represents a real brain state, but the concept is imaginary – like Santa Claus…

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  8. The author wrote:

    “Nowadays almost every critic of literature will argue that a novel has as many meanings as there are interpreters. Most of them will even argue that a novel has, in fact, potentially infinite meanings.”

    ————————————–

    This view, in fact, is not nearly as prevalent as one might think. I highly recommend the following, by Arthur Danto, probably the best and most influential philosopher of art and art criticism, in the Analytic tradition:

    “Deep Interpretation,” in The Philosophical Disenfranchisement of Art (New York: Columbia University Press, 1984).

    In this essay, Danto maintains that legitimate interpretations — what he calls “surface interpretations” — are factual ; they are based either on what the artist intended or on what he/she *could* have intended, as determined against a number of contextual facts.

    If we are talking, strictly, about art critics, probably the most influential work of the last 50 years or so, on the subject of interpretation, is Susan Sontag’s, “Against Interpretation.” There, she not only argues against the “infinite interpretations” view, but argues against the very act of interpretation itself.

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  9. @Liam Ubert

    >The universe is real and apparently it is finite

    Where do you get this from, Liam? Sure, the observable universe is finite — it has to be for a number of reasons (because the speed of light is finite, a finite time has elapsed since the Big Bang and the universe is expanding faster than the speed of light, so there is not enough time for light from the most distant parts of the universe to reach us).

    There is absolutely no evidence, however, that space does not continue infinitely in all directions. Indeed all the evidence we can amass points to it being at least vastly bigger than the observable universe and is consistent with it being infinite. If you add to this the fact that many cosmological models predict an infinite space, and the fact that the idea is so philosophically satisfying to certain infinity-sympathisers like myself, it strikes me as strange that you would describe the universe as apparently finite.

    You can certainly advance the hypothesis that there are no actual infinities, but you are on shakey ground as soon as you assert that this is the case.

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  10. For the curious who may be wondering (or worrying) about the continuum hypothesis (CH) being neither provable nor disprovable, here’s more to wonder (or worry) about. It is indeed the case that CH can be neither proved nor disproved, providing you limit yourself to particular theories of sets (notably Zermelo–Fraenkel set theory with the axiom of choice, [ZFC] in the case of Gödel & Cohen’s proofs). In one sense this is not much of a limitation, because ZFC is so powerful that it can be used to formalise the vast majority of what we generally regard as mathematics.

    However, to Platonists, such as Gödel himself, this does not necessarily mean that the continuum hypothesis is neither true nor false, but rather only that it cannot be proved or disproved with current tools. A Platonist may claim that with a better understanding of sets and numbers, the fog will clear, and we will then know whether there exists a set bigger than the rationals but smaller than the reals, (accepting that we may have to modify our understanding of numbers to know what we are talking about). Indeed, such a theory may emerge reasonably soon. The mathematician Hugh Woodin is working on an intuitively plausible theory of large cardinals called “Ultimate L”, which was partly anticipated by Gödel himself, and in which the continuum hypothesis is true.

    The truth of the continuum hypothesis would not contradict the results of Gödel & Cohen. You could still build a consistent formal theory in which the continuum hypothesis is false (providing ZFC is consistent, which it is widely believed to be), but if Woodin is correct and CH is true, such a theory, while interesting, would be no use for investigating the properties of numbers as we intuitively know them. Perhaps it would be useful for something else. Much is made of the philosophical ramifications of such situations, but personally I think there’s no need to panic. To take an example from elsewhere in maths, Euclidean geometry is a perfectly good theory of geometry, but it’s no use for doing geometry on a sphere.

    The whole debate quickly leads to the even more fundamental question of whether Platonism is the correct view of mathematics. Without wishing to come down firmly on either side of that particular debate generally, let me offer the following view: whether or not I accept Platonism (yes, I’m still worrying about it), because I think it is “true” that 1+1=2 (and that arithmetic is consistent, and some other basic mathematical truths), I expect that CH is either true or false insofar as it applies to sets of numbers (and hence to what I intuitively think of as sets), irrespective of, and consistent with, the fact that CH is neither provable nor disprovable in ZFC.

    As Gödel himself wrote in an oft-quoted passage: “…new mathematical intuitions leading to a decision of such problems as Cantor’s continuum hypothesis are perfectly possible.”

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  11. Its always nice to read imaginative accounts of nature, including whether infinities exist, but it is best to confine yourself to nature where possible, rather than abstracts. Disagreeable Me has tried to do that by referring to the likely reality of a finite expanding universe (specific quantity of mass) that might be expanding in an infinite space. This depends on scientific rigor to determine whether infinite space is still viable following the intact expanding space theories of Relativity, where space expands with mass and so might not be infinite (along with mass) but it might expand infinitely (which introduces the same issue of “infinity”)..

    I wouldn’t bother too much with abstract math, just apply the idea of infinity to the real world and to challenge Relativity. We have “some” rigor in analysis of Relativity, but it breaks down at a Big Bang, and so it is insecure, currently. Just accept that mass if finite, and explore whether space is infinite, and then deal with the strange theories of Quantum Mechanics that mass simply appears from literally nothing. Nothing in that sense is as problematic as infinity. Work with actual relevant theory rather than abstracts, its more interesting for the reader, and more relevant to everyone.

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  12. @John Smith

    > Disagreeable Me has tried to do that by referring to the likely reality of a finite expanding universe (specific quantity of mass) that might be expanding in an infinite space.

    Sorry, but that is not at all the picture I was trying to paint. What I am describing is consistent with the best cosmological models we have, and relates to the possibility that there is an infinite expanding universe with an infinite quantity of mass across infinite space. Space itself is expanding. It’s not expanding into anything. The expansion has the effect not of increasing the volume of space (which on this view is already infinite and always was) but of decreasing its density as everything flies away from everything else.

    Again, it’s just one possible picture, but the one you inferred is not taken seriously by modern cosmologists, implying as it does that the universe has a center and a periphery beyond which there is nothing.

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  13. @Tim Wilkinson

    >However, to Platonists, such as Gödel himself, this does not necessarily mean that the continuum hypothesis is neither true nor false, but rather only that it cannot be proved or disproved with current tools. A Platonist may claim that with a better understanding of sets and numbers, the fog will clear,

    I just want to note that there are more than one type of Platonist. I count myself a Platonist, but not of this sort. I am a full-blooded or plenitudinous Platonist. I don’t think there is any fact of the matter about whether mathematical axioms are true or false. There are only consistent and inconsistent systems. If it is possible for consistent systems to be built with both the continuum hypothesis and its negation, then it is true in some systems and false in others. As a plenitudinous Platonist, I believe this would imply that both sets of systems actually exist and neither is any more real than the other. All consistent systems are valid and all mathematical objects defined within consistent systems exist.

    Of course this does not preclude the possibility that the question of the continuum hypothesis might one day be definitively settled one way or the other, but that just means that one or other of the systems is inconsistent.

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  14. @ Disagreeable.

    You make some interesting points, but they sort of agree with my position which is that our “universe is real and apparently it is finite”. You say “Sure, the observable universe is finite”. I say what there is beyond the observable universe is entirely speculative. We suspect that at least some of it is real.

    Would it be correct to say that there are no infinities on earth, in the Milky Way or in the observable universe? Since the Big Bang started things at a finite point in space and time, there should be an outward boundary which would represent the limit of our real universe. Other universes or empty space could be infinite, but these are theoretical mathematical abstractions and probably will never be shown to be real. If am wrong I could certainly adjust.

    The OP suggests that these otherworldly non-real concepts of infinities will improve our practices of criticism and philosophy. I agree they enrich our lives and could improve our thinking, but there is (almost) an infinitude of things that could do that. I just don’t like nothing or infinity.

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  15. What is mathematics?

    Pieces of Logos made explicit by so-called mathematicians. They do not have to search very far: our brains are made of mathematics, entangled in all ways.

    Official mathematician? Someone doing what previous mathematicians recognize as mathematics.

    Official “mathematics” is pretty much a question of opinion. A recent example: for the longest time, category theory was not recognized as mathematics, but as “abstract non-sense”. Seventy years later, now that it delivers general powerful results, it has become respected.

    The oldest example? Euclid excluded enormous parts of Greek mathematics (non-Euclidean geometry and a modern number system), on the ground that these parts were not rigorous enough (the inchoating Greek algebra was exported to India, where it got developed fully, and then re-imported by the most famous Persian mathematicians in the Ninth Century).

    In my opinion, the entire theory of infinity in mathematics is erroneous. Don’t laugh: although Euclid’s propositions were not properly speaking erroneous, it deduced propositions which he did not prove. For example, there is no Euclidean axiom saying two circles of the same radius whose centers are less than a radius apart, intersect. And that proposition is not a conclusion from Euclid’s axioms.
    OK, it is “physically” “obvious”.

    Is unproven mathematics still mathematics? Is mathematics proven with reasons not discussed really proven?

    And it is just as physically obvious to me that mathematical infinity cannot exist.

    Why? It is not just that the observed universe is finite. It is also that it contains a finite number of particles, as an elementary application of the Planck’s Quantum emission law shows. Why would there be more numbers than particles? Surely, you are joking, Mr. Mathematician?
    http://patriceayme.wordpress.com/2011/10/10/largest-number/

    Don’t scoff: there is precedent of sorts. The famous Dutch mathematician Brouwer developed in the 1920s “Intuitionistic Mathematics”.
    http://plato.stanford.edu/entries/intuitionism/

    This brought a severe conflict within mathematics. A star mathematician (there is something called the Brouwer Fixed Point Theorem, which is most useful), at the top of the profession claimed that the Foundation Program (Russell, Hilbert, etc.) made no sense, for a question of, well, intellectual honesty.
    Brouwer got very disliked, he was kicked out of Annals of Mathematics, and mathematicians scoffed.

    However, starting in the 1970s, all automatic tellers use Brouwer’s ideas. So Brouwer won.

    In truth, there are no serious foundations to mathematics. So much for them being pure ontology. Mathematics is all about pieces of Logos, like castles floating in the air. When they work, that’s great. How do we know they work? Well, it’s pretty much an experimental endeavor.

    For example the lack of a resolution of the Navier-Stokes equation means, in practice, that mathematics cannot be used to compute hypersonic flow (thus engineers just go out and experiment), or cavitation around propellers.

    If mathematical infinity cannot exist, what changes? Nothing. Computers use finite mathematics.
    https://patriceayme.wordpress.com/2013/10/31/finite-calculus/

    The Continuum Hypothesis is that there is NO set whose cardinality is strictly between that of the integers and that the “continuum” of the real numbers. There is only one countable infinity.

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  16. To ask Disagreeable:

    “..the question of the continuum hypothesis might one day be definitively settled one way or the other, but that just means that one or other of the systems is inconsistent.”

    Godel proved that if the system ZFC of axioms is consistent, then the system which is ZFC plus one more axiom, namely the continuum hypothesis, is also consistent.

    Cohen proved that if the system ZFC of axioms is consistent, then the system which is ZFC plus one more axiom, namely the negated continuum hypothesis, is also consistent.

    So I cannot understand what you meant by the quoted sentence. Can you explain? Thanks.

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  17. @Bill Skaggs
    Bill Skaggs correctly points out that the mathematical term of “infinity” is use “far too loosely” in humanities. And I think that is true, but we should also remember it is such because the mathematical sense of infinity can only be use rigidly when dealing with mathematical systems. As far as I understand, in mathematics the term infinity is used to indicate a measure of quantitative value, such as the number of elements of a set; on the other hand, in humanities when we talk about the infinity we usually are using it as a kind of measure of value of intangible things; such as the value of an interpretation, or of a piece of art, or the value of a person’s live.
    The meaning of “infinity” is clearly different in both math and humanities. The same goes when talking about the infinity in physics. The point I’m trying to make is that although both humanities and mathematics (and physics) deal with different terms of infinities, both “infinities” are analogous to each other. -By the way, by an analogy or metaphor (I’m using these terms as synonyms here) I mean a type of thinking that makes emphasis in the similarities between two distinct things, putting the differences assay-. As an analogy always compares two different things, I think it is one good instrument to mediate between different kinds of knowledge. I also think that a basic understanding of the mathematical term of infinity is helpful when drawing analogies in other fields of inquiry.

    @Phil Thrifty, @Liam Ubert @Disagreeble Me
    I remain skeptic in the hole topic about the infinity of not of the universe or the hypothetical multiverse. I just have so many problems grasping the idea of the limits of the universe. Just ¿What does it mean to talk about infinite times and spaces?
    But I think that the matter of infinity physically existing or not –whatever that means- doesn’t change the humanistic notion of infinity as explained above. Although it is certainly thrilling to think about it.

    @Darko Mulej, @Disagreeble Me
    I find both of your analogies very interesting and enriching. Personally, I think that the core of the question here is whether we consider or not every layman’s interpretations and a critic’s interpretation as fundamentally different or not.

    @Aravis Tarkheena
    Thanks for pointing that out. I’ll try to read about Susan’s and Danto question to see if I can comment something.

    @Tim Wilkinson
    That’s very interesting information. You may like to read more about Badiou’s philosophical program named “Platonism of the multiple”.

    Like

  18. Perhaps I’m wrong, but I think the main purpose in this essay is to make an analogy between mathematical infinity and literary infinity in order to suggest that literary critics (or people who aspire to be one) can learn something from mathematical infinity. My problem is that the analogy is too loose to say anything in substance. The author did concede his analogy is just an analogy. For example:

    “Of course, this is an analogy. Philosophy and math are based on different kinds of reasoning, and I’m pretty confident they will stay this way forever.”

    But my concern is that the author doesn’t seem to be aware that analogies have several purposes (or that there are different kinds of analogy). The most popular use of analogy is to help convey very difficult concept/phenomenon by comparing it to another concept/phenomenon that we already grasp. This is interesting for pedagogical purposes, but that’s not the kind of analogy that’s philosophically interesting. A philosophically interesting kind of analogy is when you inductively compare two phenomena by going over series of properties and then conclude that if A has X property, then Y is likely to have X property too. Or, in science, analogies are interesting when you see common patterns among two apparently disparate phenomena. So, which kind of analogy is he using exactly?

    It seems to me that the author is using the analogy to illuminate how literary critics are still relevant despite that lay people can may so many interpretations of the same literary work. He’s making analogy between integer/real numbers and lay interpretation/well-reflected interpretation. But the analogy he makes is that laypeople can make more interpretations that are easy to make as oppose to literary critics who can makes less but well informed interpretations. Just as there are more real numbers than integers, there are more lay interpretations than well-reflected ones. But I think the author misses another distinction: weak vs. strong analogy. On one hand, a weak analogy points out superficially similar properties of two or more phenomena, but those properties aren’t very pertinent in developing any deep understanding of those phenomena. On the other hand, a strong analogy points out deep similarities between two or more phenomena by pointing out essential properties they have in common; properties that are pertinent to developing some deep understanding of those phenomena.

    The kind of analogy that the author is making doesn’t come across as a strong analogy. He doesn’t seem to point out any important similarities between literary and mathematical infinities to come to some very interesting and insightful conclusions about their relation to one another. This comes across as a weak analogy insofar as it points out superficial properties: just as there are more lay interpretations than well-reflected interpretations in literature, there’s more real numbers than integers. Similarities seem superficial because it’s showing similarities in terms of loose quantitative ratio. However, even if I charitably grant that the weak analogy is at least mildly interesting, I still think there’s a point where it breaks down. In the first part of the analogy (literary infinity) the author is making an evaluative claim about lay interpretation vs. well-reflected interpretation; there’s more lay interpretation because its based on *quick and unreliable conjectures, whereas well-reflected interpretations have less but more substantive/informed interpretation. After all, interpretation is inherently a normative/evaluative enterprise. Mathematical infinities, on the other hand, aren’t necessarily normative/evaluative, but more or less descriptive.

    I’m still not sure what literary critics are suppose to learn from mathematical infinities based on the author’s analysis. Perhaps the author’s analysis is up to infinite interpretations, but I’m not sure if there’s even the best one.

    Like

  19. Since DM may be running out of comments:

    Hi Liam Ubert,

    Since the Big Bang started things at a finite point in space and time, there should be an outward boundary which would represent the limit of our real universe.

    The standard Big Bang model does not start with a “finite point”, it starts with an infinite extent of stuff around about the Planck time. Thus, in the standard cosmological model our “real” universe is indeed infinite in extent.

    Multiverse models do indeed have our universe starting with a finite extent, but as a bubble in a pre-existing universe, where the ensemble would again be taken as of infinite extent. Thus if you restrict “our real universe” to one bubble of a multiverse then you’re right that it is finite, but beyond that is more stuff.

    Hi phoffman56,

    It is important to distinguish between infinitely many sentences and infinitely lengthy sentences. The latter is pretty much excluded, if not theoretically out of the question. But the former is clearly true of English, as in the example: I am. I am and I am. I am and I am and I am. …etc….”

    If you have a limit to the length of a sentence, and a finite number of different English words, then the number of combinations of those words (within the maximum length) is finite.

    Like

  20. This is my last comment, so hopefully I can wrap up a few of the points outstanding.

    @Liam Ubert

    I agree with Coel’s response (and thanks Coel, by the way, please feel free to continue answering points on my behalf after this my last comment).

    > Would it be correct to say that there are no infinities on earth, in the Milky Way or in the observable universe?

    It is currently unknown. There may be singularities in the centres of black holes (e.g at the centre of the galaxy). It rather depends on how quantum mechanics is unified with general relativity.

    > Since the Big Bang started things at a finite point in space and time, there should be an outward boundary which would represent the limit of our real universe.

    As Coel has said, this is not a correct understanding of how the Big Bang should be interpreted if the curvature of space is zero as it appears to be (or if the curvature is negative).

    @phoffman

    >So I cannot understand what you meant by the quoted sentence. Can you explain? Thanks.

    I think this is probably just me being dumb. I felt insufficiently confident to assume that the question of the CH could never be settled in some way. Tim Wilkinson said:

    “A Platonist may claim that with a better understanding of sets and numbers, the fog will clear, and we will then know whether there exists a set bigger than the rationals but smaller than the reals, (accepting that we may have to modify our understanding of numbers to know what we are talking about). Indeed, such a theory may emerge reasonably soon. The mathematician Hugh Woodin is working on an intuitively plausible theory of large cardinals called “Ultimate L”, which was partly anticipated by Gödel himself, and in which the continuum hypothesis is true.”

    I’m not in a position to judge if that kind of approach might count as some kind of resolution to the question. I’d be interested in your taking up the point with Tim.

    If it is assumed that mathematics is consistent with both the affirmation and the negation of the continuum hypothesis, then you can take my point to be a counterfactual example to illustrate that full-blooded Platonism should not be interpreted as the view that all mathematical systems exist, only that those which are consistent exist. Of course we can’t generally prove which systems those are, so the assertion is that existence is equivalent to consistency. We can’t be sure what exists because we can’t be sure what is consistent.

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  21. Hi Coel,

    Thanks for the reply:

    ‘..Hi phoffman56,

    “It is important to distinguish between infinitely many sentences and infinitely lengthy sentences. The latter is pretty much excluded, if not theoretically out of the question. But the former is clearly true of English, as in the example: I am. I am and I am. I am and I am and I am. …etc….”

    If you have a limit to the length of a sentence, and a finite number of different English words, then the number of combinations of those words (within the maximum length) is finite…’

    I’d followed what you quoted from me above by

    “..This requires the existence of sentences of arbitrary, but finite, length…”

    This was my less clear way of formulating what you added above more clearly (I think!)

    Separately, I think I was a bit brief with the later thing that DM said and both of you partially qualified. He may have been thinking of the following, which I believe is in general what Woodin is attempting, and what Godel had alluded to as a possibility (though I think Godel predicted that CH is more likely false than true, if something like the following came to pass):

    Somebody discovers a formula R in the language of 1st order set theory, a formula expressing a proposition which all reasonable people (not necessarily all Australian logicians) believe to be intuitively true about ‘real sets’ (and also maybe not hidebound finitist philosophers here). It is also shown, in analogy to the Godel and Cohen famous results that:

    If ZFC is consistent, then ZFC plus R as an added axiom is consistent.

    And even more, it is shown that CH follows in the system ZFC plus R, say, by Woodin. Then it is reasonable for those reasonable people to accept the continuum hypothesis as true.

    Or perhaps the ‘opposite’, i.e. replace CH by its negation just above, to confirm my recollection above of Godel’s possible guess.

    Like

  22. @ Philonous13

    That’s a valuable distinction to make between different kinds of analogies. I would like to make two comments.

    First off all, I agree with you in that the analogy between “integer/real numbers and lay interpretation/well-reflected interpretation” is indeed a week one. But the porpoise of this specific interpretation is to encourage debate and see if something interesting could arrive from the discussion. So far some interesting suggestions have aroused.

    On the other hand, I do think there’s a strong analogy in the text, and is addressed in the first part of the essay. The idea is that the infinity in both disciplines should be understood as multiple instead of absolute. As far as I know this idea was first coined by Alain Badiou:

    “Mathematics […] has treated the infinity with the triviality of the cardinal number. Has neutralized and desacralize the infinity […] tearing it apart from the realm of the One, to disseminate it in the secular typology of the multiplicity. By founding a way of thinking in which the infinity is irreversible cut out from the circle of the One, mathematics has really accomplished, by its own account, the program of the death of God”. [Translated by me form the Spanish version of “Conditions”].

    I would say that this strong analogy is the main point of the essay. Also it is to encourage people in humanities to learn some mathematics, because in my personal experience (I don’t know how representative this observation is outside my faculty) there’s a big stigma and misunderstanding of mathematics and even science in humanities, especially in cultural anthropology and literature. For example, Bruno Latour, Foucalt and Derrida are very popular, just to name a few.

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  23. “According to Alain Badiou, the history of Western philosophy can be divided into two great periods. First, the era before and including Kant, when mathematical reasoning was considered a singular way of thinking that interrupted the predominance of opinion — or, to put it in philosophical jargon, of Doxa — in philosophical reasoning”.

    I’m doubtful on this point. Plato’s idealistic and mathematical theory was criticised by the Peripatos with physic and logic reasons. The Cynics issued an existential philosophy in which mathematical reasoning wasn’t relevant, and the Stoa tried an ontological inquiry that matched whit logic or at least was compatible. Parmenides didn’t bother about mathematics to show the Self, and his theory has survived until now.

    “And second, the post-Kant era, which gave birth to Romanticism, which was consummated by Hegel, whose philosophical system is powered at its core by the schism between math and philosophy.”

    I wouldn’t say that Hegel was romantic, actually he was phenomenologist and was skeptic about Aristotle’s episteme. Hegel feared that the empirical knowledge could become tautological and circular, that’s why issued a meta-logic discourse to avoid such circularity. On the other hand, he made an analysis of infinitesimal calculus that didn’t solely focus in applying the formal rules of signification because he thought that such rules hidden the true meaning of these operations. Thus, I don’t see a schism regarding Hegel.

    “In fact, for Hegel, nature is death Logos, or ideas, and as such it has no real value for human understanding”.

    I don’t understand this statement. Logos is not synonymous of ideas, according to the Stoa Logos is a rational principle underlying the functioning of the universe. Fortunately or unfortunately there are several views on this topic, namely, there are different ideas to explain what Logos is.

    “This idea he called the hypothesis of the continuum [CH], and he could never prove it. In 1938 Kurt Gödel proved that it is actually impossible to show that this hypothesis is false; and, even more amazingly, in 1963 Paul Cohen proved that it is also impossible to prove it correct! So we are condemned to live with this ignorance forever!”

    Well, Kurt Gödel added or inserted in the CH the axiom of choice and proved that the CH wasn’t contradictory with Cantor’s set theory. Gödel proved that the negation of the axiom of choice is contradictory with ZF, therefore this axiom is consistent. Assuming that ZFC is consistent Paul Cohen used the technique of forcing, developed for this purpose, to also prove that ZFC is consistent.

    The generalized continuum hypothesis, GCH, seems independent of ZFC but, amazingly, the Polish W. Sierpiński proved that ZF + GCH entail the axiom of choice, so choice and GCH are not independent in ZF. Does it mean that Cantor was right? He believed that the continuum hypothesis was true and tried for many years to prove it, though in vain.

    Cheers

    Like

  24. Hi Jorge, thank you for posting, if you purpose was to stir up discussion it looks like you have succeeded.
    I wanted to press you a bit more on Philonous’ concerns because I don’t think his worries were adequately responded to.

    First- Philonous was concerned that it was unclear whether or not the analogy you draw in your paper between integer/real numbers and lay interpretation/well-reflected interpretation was merely a pedagogical analogy: an analogy that helps someone understand concept A better by juxtaposing it with concept B, but not one that makes a claim about A by saying A is similar to B. For example (though I have no convictions on this matter) people have asked me, “how could god = the father, the son, and the holy ghost (the holy trinity)?” The reply I tend to give is, “well perhaps people think of this identity as similar to a proton and the quarks that compose it. A proton is made of 3 quarks, each quark has its own individual properties, but the proton (the 3 quarks bound together in certain ways) has its own properties over and above the individual quarks.” This analogy is just useful for understanding how one entity, god, could be identical to 3 entities. Similarly, Philonous was wondering if all you were doing in your essay was helping people understand how a literary work can have infinite critical interpretations, while at the same time not all its interpretations are critical.

    If that is all your analogy was doing, then Philonous (I think) felt as though this is a pretty weak and not terribly insightful thesis (though it certainly could still be used as a pretext for discussion).

    Second- Philonous was concerned that if you weren’t just making a pedagogical analgogy and were actually trying to make a philosophical analogy (saying something like, “concept A has property Y, concept B is like concept A, concept B has property Y) then there might be further concerns:

    1- the integer/real numbers and lay interpretation/well-reflected interpretation seems like a weak analogy (only saying something superficial about the two things which isn’t terribly informative). This is something you agreed to above.
    2- If you think that there is a strong analogy in the text, which you say there is, we need to know why it is a strong analogy. You say, “On the other hand, I do think there’s a strong analogy in the text, and is addressed in the first part of the essay. The idea is that the infinity in both disciplines should be understood as multiple instead of absolute. As far as I know this idea was first coined by Alain Badiou:

    “Mathematics […] has treated the infinity with the triviality of the cardinal number. Has neutralized and desacralize the infinity […] tearing it apart from the realm of the One, to disseminate it in the secular typology of the multiplicity. By founding a way of thinking in which the infinity is irreversible cut out from the circle of the One, mathematics has really accomplished, by its own account, the program of the death of God”. [Translated by me form the Spanish version of “Conditions”].

    I would say that this strong analogy is the main point of the essay.”

    There are two problems with this:
    1- if it is a strong analogy, we need to know why because we haven’t been given a reason to believe it is from your above quote. You have just asserted that it is a strong analogy without further support. In fact, this really still looks like a pedagogical analogy. Understanding infinity in both disciplines as multiple instead of absolute seems to just be another way to understand the answer to your friends’ question from the beginning of your essay (how could a literary work have infinite critical interpretations, while at the same time not all its interpretations are critical?).
    2- if it is a strong analogy telling us something deep, Philonous was concerned that mathematics and literature are not similar enough to warrant using your analogy. He pointed out that in literature interpretations, we are paradigmatically making evaluative/normative claims about one interpretation being better than the other, whereas mathematics is just a descriptive system devoid of normativity. This seems to Philonous to be a deep difference between the two things in your analogy. This difference would render your strong analogy pretty unreliable.

    Once again thank you for contributing your essay on this post. I enjoyed reading your paper but would like for you to say more about Philonous’ concerns if you have the time.

    Like

  25. Also instead of simply pressing you on Philonous’ objections, I wanted to compliment you on your clear exposition of how Cantor proved various relations between infinite sets. I am sure there are many people who didn’t understand these relations and you have made the material very accessible.

    I also appreciate that you saw a need for understanding on a certain issue in ordinary dialogue with friends, and decided to try to convey understanding to the general public as a result. I think this is admirable. Congratulations.

    Like

  26. Re Daniel Tippens’s comment: Thank you! Shout out to Philonous, of course, and to Aravis as well.

    Like

  27. Disagreeable Me, you now say mass and space are both infinite and that accords with the “best” theories. At least that is now clear.

    Liam Hubert said “I say what there is beyond the observable universe is entirely speculative”, and I agree with that when it comes to the kind of speculation you propose. I don’t care whether the universe I propose is taken seriously by modern cosmology if modern cosmology is as you say. I don’t actually think modern cosmology is as you say, but that is moot anyway because modern cosmology completely breaks down at your original “creation event”. However, I can still deal with what you say.

    So this is an “infinity” that expands, and this includes an infinite mass – “expansion has the effect not of increasing the volume of space (which on this view is already infinite and always was) but of decreasing its density as everything flies away from everything else.” Infinity can be more or less dense and remain infinite, and this includes infinite mass? An abstract use of terms that I reads as unintelligible.

    Maybe with “continuous creation” theories you can have mass continuing to be created “infinitely”, but the better view held by cosmologists is that you could actually count a finite number of particles, and there is a finite quantity of mass in the universe that expanded at a Big Bang. Finite does not mean infinite, even if it expands infinitely as a finite mass.

    I can see how you got stuck with an infinite mass if you propose an infinite space as an original state that decompresses along with it – no choice, boxed in. Its illogical in terms and just a speculative view interpretation of reality. No need to say sorry about the confusion in your post, just work on secure definitions.

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  28. Coel wrote: “Thus if you restrict “our real universe” to one bubble of a multiverse then you’re right that it is finite, but beyond that is more stuff.”

    Disagreeable wrote: “It is currently unknown [if there any infinities in the observable universe]. There may be singularities in the centres of black holes (e.g. at the centre of the galaxy). It rather depends on how quantum mechanics is unified with general relativity.”

    This is reassuring because I do intend to restrict myself to the observable universe; even more than that, I intend to continue to concentrate on the bioverse. So, ordinary or common sense logic should apply to human affairs. Deep speculations in terms of maths, physics and philosophy can be beautiful, thrilling and inspiring, and can greatly contribute to our human existence. But that is certainly not a sure thing.

    The OP suggests that it would be of benefit for the humanities to adopt concepts from other disciplines. This kind of intuitive cross-pollination might be an essential strategy by which one can obtain a general understanding of the frameworks of our human existence, super duper complex as it is. Our culture is so complex that we might as well say it is infinitely complex, even though that would not be strictly correct in our finite observable universe. However, using the weird logic of mathematics, and submitting to the imperatives of scale, functionality and practicality, it appears that our existence is infinitely complex. As finite human beings we cannot become the master of existence, we can only be the captain of our own life raft. The way in which humanity deals with this ‘irreducible complexity’ is through a division of labor: science, philosophy, art, math, farming, production, capital, administration, etc, ad infinitum – each often with their own language. Some may specialize in being generalists and try to integrate all this information. This would be the most difficult specialization of all, impossible really.

    This brings us to another point in the OP: “The main difference between real interpretations and integer interpretations of a literary work is that the first ones are more numerous, easier to produce and, hence, less valuable than the second ones.” This is very questionable given that all humans are severely handicapped by their limited knowledge of our infinitely complex culture, with NO ONE having a coherent picture of the whole. It is therefore extremely hazardous to rely on professionals for the inside scoop. Each one of us is in charge of and responsible for their own universe, of what comes in and what goes out of our so-called idioverse. One can certainly expect a professional to provide better quality information, but that too often does not happen. To rely on a professional for an interpretation as it applies to your own existence is even more risky, especially if it goes beyond the narrow field of their concentration. But sometimes one must. I would suggest that there are far more amateur experts in literature than reliable professional critics. A useful discipline is to first form one’s own opinions on how to solve a problem and then check to see what others might have noticed.

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  29. Lim: “Some may specialize in being generalists and try to integrate all this information. This would be the most difficult specialization of all, impossible really.” You could start with this bold attempt and see what you think, but it took me ages to get the drift, very new ideas http://1drv.ms/1tnKM6f

    Like

  30. Even if “lay” people and trained critics uttered one million words per second, in the end, all they would have said, and will say in the next century is finite. So I don’t see what the infinite has to do with the subject at hand.

    This being said, I like to see the “infinite” appear in a conversation. So I salute the courageous effort of the author. I will be less kind to Badiou, who has sometimes criticize others with the elegance of a hungry crocodile.

    “The idea is that the infinity in both disciplines should be understood as multiple instead of absolute. As far as I know this idea was first coined by Alain Badiou:
    Mathematics […] has treated the infinity with the triviality of the cardinal number. Has neutralized and desacralized the infinity …”

    I don’t think Badiou knows enough mathematics to suspect how ludicrous he sounds to a mathematician.
    Mathematicians tried to understand the concept of infinite, or infinitesimal, in two ways. The first was with Leibnitz. After his death, Bishop Berkeley made fun of Leibnitz’s “infinitesimals”. Yet, they worked.
    It took three centuries to make infinitesimals rigorous, through a branch of Logic called Model Theory. That solution, Non-Standard Analysis, is philosophically fascinating, and it has the same smell as Category Theory, namely a subtle game with axioms.

    Category Theory builds out of thin air. Non-Standard Numbers are created by refusing Archimedes Axiom.

    The second attempt with infinity was with Cantor. And it rules. Cantor added the concepts of bijection, injection, surjection. Two sets have the same cardinality if, and only if, they are in bijection. Cantor also invented a number of tricks, the most famous being his diagonalization (which has been used again and again, and again by diverse famous authors).

    Cantor’s work actually rests on the Successor Axiom of Euclid. It’s viewed as one of the foundations of mathematics. Yet, after careful consideration, I refuse it. That refusal is the opposite, in Model Theory, of the Non Standard construction (which adds infinities absolutely, whereas I cancel them, just as absolutely).

    Morality? In the fullness of time, mathematicians have tried a lot of things. Good philosophers have to be aware of what they tried, before criticizing it. Otherwise they will sound like a bleating lambs.

    Like

  31. Mario Roy,
    ‘“In fact, for Hegel, nature is death Logos, or ideas, and as such it has no real value for human understanding”.
    I don’t understand this statement. Logos is not synonymous of ideas (…).’

    For Hegel (in this context, inorganic materiality) is pure negativity. It is therefore ‘dead,’ inanimate and incapable of animism. Therefore any concept derived from it or concerning it cannot be The Idea, that which we seek to know. such concepts can only be ‘ideas’ in the loose sense. We ‘know’ the rock, but we can only ‘know’ its physicality, we cannot know what ‘rockness’ means until we have sublated it into our conceptualization of the phenomenon. This is inherently sterile and facile knowledge, and does not effect our knowledge as knowing subjects, until we recognize it as a negation of our subjectivity (‘thatness’ without incorporation), that must be sublimated into our subjectivity as a higher order conceptualization (what is this rockness and what does it mean to me as the knower of any rockness at all?).

    The Logos is a loose concept; the stoic conceptualization is one perspective, but not the only such. In Hegelian terms, the reading of Logos as Stoa is one sided – matter infused with Spirit (matter conceptualized by knowers in its totality). The sublation is clearly matter-and-spirit as one (Spinoza), which begs the question, what is the Spirit that sublates this as well to achieve self-realization? (Which Hegel claims to have supplied.)

    This actually has much to do with the OP’s main discussion. The ‘false infinity,’ whether in maths or physics, is a mechanical repetition, and thus merely duplication of inorganic matter. The true infinity is a stage toward the Spirit’s self-realization as Idea.

    Like

  32. I am not aware of any scientific evidence that there are any infinite things in nature. My claim is if there were, we could make working hypercomputers. For now these only exist in science fiction and theory.

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  33. The term “infinity” means different things in different contexts.

    In everyday/colloquial use and in the humanities it simply means “very large”. In the spirit of the counting scheme that goes “one, two, three, many”, the word “infinity” is most often used for things that aren’t infinite but just large. Indeed, Oxford Dictionaries allows meanings such as: a “very great number or amount”or “impossible to measure or calculate”.

    In physics the term has a more specific meaning, it means “so large that any upper bound has no observable affect and thus can be neglected”. Thus a cosmological model that extends “to infinity” really means it extends sufficiently beyond the observable horizon that any “edge effects” can be neglected. Extending a model to infinity is thus a way of taking the simplest version of the model (in line with Occam’s razor).

    The mathematicians, of course, take a stricter and more literal approach, and consider actually infinite sets.

    John Smith,

    Infinity can be more or less dense and remain infinite, and this includes infinite mass? An abstract use of terms that I reads as unintelligible. […] the better view held by cosmologists is that … there is a finite quantity of mass in the universe that expanded at a Big Bang.

    DM’s comment that you were replying to was a correct account of current cosmology. As above, the standard model extends “to infinity” at the start of the Big Bang. Thus it starts with infinite extent and infinite mass. The reason for this is that there is no evidence of any upper bound (either on the mass or on the spatial extent) and thus these are taken to run sufficiently beyond any upper bound that there would be no observational consequence of any upper bound.

    (Of course one can then consider multiverse variants of the Big Bang in which you do have a bound to the region undergoing primal inflation, since beyond that it transitions to the pre-existing, non-inflationary state in which the Big-Bang bubble started. As yet, there is no convincing observational evidence of such an “edge”, and thus, as yet, the standard model just extends to infinity.)

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  34. Ejwinner,

    I don’t understand your comment, I just remark that Hegel feared to focus on the empirical knowledge our understanding of nature. Whether Hegel was right or not is beyond my knowledge. Hegel issued a meta-phenomelogical logic that is close to Bolzano and Husserl inquiry. Such logic doesn’t comply with “the formal rules of signification”. Thus, he afforded a deep analysis of the infinitesimal calculus to explain his view on this matter.

    I agree with Coel regarding the physic infinite. He wrote:

    “As above, the standard model extends “to infinity” at the start of the Big Bang. Thus it starts with infinite extent and infinite mass. The reason for this is that there is no evidence of any upper bound (either on the mass or on the spatial extent) and thus these are taken to run sufficiently beyond any upper bound that there would be no observational consequence of any upper bound”.

    It seems plausible that the observable universe couldn’t start from nothingness (ex nihilo), that means that there were no mass and extension limits at the beginning of our universe. This is an interesting point that suggests that what we call the Big Bang is a step inserted in a creative and unlimited process.

    So, Cantor’s attempt to state the mathematical infinity goes along with this unlimited process that owns an ambiguous starting point.

    Like

  35. Jorge Alejandro Laris Pardo: “What we don’t always realize is …, what we really want is for it to be relevant to our lives in a way that may contribute to enriching our existence. … For mathematical knowledge to turn into wisdom it is necessary to include philosophy, and literature.”

    Amen! Excellent article, many interesting points. I would however like to make one suggestion. Although integers and prime numbers are having the same ‘set size’ at the countable level, there are much less prime numbers in any given ‘finite interval’. So, the Prime-interpretations could be the better ones among all. In fact, this is the vital point on this issue. There are ‘differences’ among countable.

    Then most importantly, there are ‘differences’ among infinities, and this is ‘the’ key for ‘this’ universe. It is obvious (needs no proof) that all infinities are ‘immutable’. Then, how can we (or themselves) transform one infinity to a different one? By knowing the answer for this question, we will know all mysteries, let alone to say about the issue of the ‘number’ types of infinities.

    As the hypothesis of the continuum is unresolved after 150 years, it will never be resolved in the current ‘fields’. That is, the answer must be in a new ‘field’ which must be connected to those current fields via a ‘secret passage way’. Of course, all this talk is nonsense unless we can show the secret garden. I will show this with two steps.
    One, construct the ‘concepts’ philosophically.
    Two, show the manifestations of those concepts with math equations and solid physics-processes.

    Concept 1, ‘Physics-ism (not physical-ism)’: the true TOE (theory of Everything) which consists of a Triplet.
    Physics 1 (P1): Expression 1 = physical universe {prequark, quark, (baryon, lepton), atoms, … galaxies; DNA… etc.}
    Physics 2 (P2): Expression 2 = bio-‘attributes (not functions)’ {intelligence, consciousness, mortality, spirituality, etc.}
    Physics 3 (P3): Expression 3 = Math {zero, finite numbers, infinities}

    Concept 2, Physics-ism: although each of the triplet has different ‘body plan and physiology’, they all have the totally identical DNA.

    Concept 3, the keys for their unification hinge on their ‘differences’. The easiest differences are between P1 and P3.

    In P1, there is neither ‘nothingness (0)’ (see https://scientiasalon.wordpress.com/2014/08/04/p-zombies-are-inconceivable-with-notes-on-the-idea-of-metaphysical-possibility/comment-page-3/#comment-5683 ) nor ‘infinity’. The best ‘zero’ in P1 is the Cosmology constant (with 120 zero after the decimal point, but it might have a nonzero digit after that).

    Then, what is nothingness (its expression and essences)? See https://scientiasalon.wordpress.com/2014/10/31/mark-english-on-philosophy-science-and-expertise-a-naive-reply/comment-page-2/#comment-9356 , and they are ‘timelessness and immutability’. Yet, in order to connect these to P1, they must be ‘processes’. And,
    Timeless ‘process’ > Alpha (= 1/137.0359 …)
    Immutable ‘process’ > string-unification (Standard model fermions).

    I have outlined the ‘philosophy’ and the outcomes. But, the beef is about the actual math equations and the solid physics-processes.

    Concept 4, the secret passage way is the ‘principle of unreachable’ of P3 (see https://scientiasalon.wordpress.com/2014/08/21/defending-scientism-mathematics-is-a-part-of-science/comment-page-1/#comment-6532 .

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  36. Mario Roy,
    I was just trying to explain the (admittedly oddly phrased) sentence from the OP, how Hegel could see nature as a “death Logos” – a logic lacking Spirit (i.e., telos towards unity of consciousness).

    I’m aware of Hegel’s efforts to read the calculus of infinitesimals dialectically; indeed, I’ve found an interesting reading of it (as embedded in H’s general discussion of Quantity) by David Gray Carlson (http://www.hegel.net/carlson/Carlson2001-Hegels Theory of Quantity.pdf). The reading is very difficult; but as I discovered when writing my dissertation, when one tries to interpret Hegel, one tends to write like him.

    (I hope it’s understood that I wasn’t advocating Hegel’s position, just trying to indicate what it might be.)

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  37. @Daniel Tippens
    Sorry for the delay but I went to the entrance on “Analogy and Analogical Reasoning” [1] in the Stanford Encyclopedia of Philosophy. According to it, an analogy can have a justificatory role when is offered in support of some conclusion. But the “intended degree of support for the conclusion can vary considerably”.

    “At one extreme” it gives the example of hydrodynamic analogies that “exploit mathematical similarities between the equations governing ideal fluid flow and torsional problems.” Is this what you refer as strong analogies? If that’s the case then I will have to admit that the analogies expressed in the OP are not strong analogies, since it is hardy the case that there exists an underlying structure shared by both literature and mathematics.

    “At the other extreme,” the entry continues, “an analogical argument may provide very weak support for its conclusion […] Often the point of an analogical argument is just to persuade people to take an idea seriously.” And the entry goes on to quote Darwin himself, who made this analogy between Artificial Selection and Natural Selection to argue for the possibility of the latter. It is clear that Darwin’s analogy is good to understand Natural Selection; in the other hand, if we take this analogy too far we might be tempted to think that there is some type of teleological reasoning in nature that is guiding evolution (and the last time I ask the big rock in my yard about this supposed plan of nature, it told me it wasn’t the case). Is this what you mean by pedagogical analogies? I will argue that the analogy I made is analogous to Darwin’s analogy (sounds funny). Both analogies point out some shared feature between two things that otherwise, from other perspective, would look completely different.

    In a different sense, I argue that everyone who wants to talk about infinity in whatever discipline they are dealing with should at the very least be inform about what mathematics has discover about infinity. I think this is an extension of what Massimo constantly emphasizes, that good philosophy is always inform by science (and in this case by math), although good philosophy can’t be reduce to science. In Badiou’s jargon, there are 5 conditions for philosophy math, art, love, politics, and science.

    This is to me a very important point because some philosophers have a profound misunderstanding of this specific topic. For example, Spanish thinker José María Mardones argues that “Our human reason is finite” because it can’t grasp absolutely all elements of nature [2]. Well I don’t know what does it mean to have a finite or infinite reason, but a basic understanding of math’s infinity teaches us that not having all the elements of a given set is not a sufficient condition to call something finite. Because infinity does not means Absolute. In this sense the analogy is kind of a prescriptive definition the word “infinity”.

    Finally, there’s jet another reason for the use of analogies, which has been theorized by some philosophers in hermeneutics, like the Mexican Mauricio Beuchot [3]. In his theory, analogous reasoning is a way to overcome the incommensurability problem

    “Analogy is the same as proportion […] In fact, an analogy always implies proportionality, this means, a relationship in witch something [*say, the concept of infinity*] can’t be attributed to different things without knowing what they have in common and, even more, what they have in difference. […] It implies to be conscious that the other is much more than what we can grasp and comprehend. It may be communication between one culture and other, or even within the same culture; between men and women, or even between man and man. […] Even though our dialog is limit and poor, it is always enricher. [The only condition is that we must always be conscious of the limits of an “analogy”]”.

    And this is why I argue a valid way to mediate between different kinds of knowledge is by analogical reasoning.

    And also thank you and everyone for your interest.

    [1] http://plato.stanford.edu/entries/reasoning-analogy/#PhiFouForAnaRea
    [2] & [3] http://books.google.com.mx/books?id=QUNGbH41YSsC&pg=PA143&lpg=PA143&dq=tiene+la+analog%C3%ADa+alg%C3%BAna+funci%C3%B3n+en+el+pensamiento+filos%C3%B3fico&source=bl&ots=c1OuP3umtE&sig=MZrB04rzf2LDHgIXMsetB9XcPxM&hl=es-419&sa=X&ei=yW92VLz2KcWpNruJgtgP&ved=0CB4Q6AEwAA#v=onepage&q=tiene%20la%20analog%C3%ADa%20alg%C3%BAna%20funci%C3%B3n%20en%20el%20pensamiento%20filos%C3%B3fico&f=false

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  38. @ejwinner @Mario Roy

    According to the Italian philosophers Dario Antiseri and Giovanni Reale [1] Hegel gets inspiration from the Neo-Platonism idea of remaining, procession, and return (Mone-proodos-epistrophe) and applies it to synthetize Fitche’s and Schelling’s ideas about Nature. In this way, Hegel conceives that in the origin the Logos sacrificed itself in behalf of its own benefice. The consequence of this act was the origin of Nature, which is explained as “the idea” outside of itself and is studied by Natural Philosophy. To understand this idea, let’s imagine a craftsman that makes a glass. The glass is a product of her ideas (logos), while the glass has no ideas of its own, just like nature. Nevertheless, the glass wouldn’t exist if it weren’t for the original ideas of the craftsman.

    Then, Hegel explains that the process from which The Idea returns to itself gives birth to the Spirit. So the Spirit arises from Nature and it’s the movement that thinks about itself. The Spirit is some kind of improved version of the Logos and it’s the object of study of the Philosophy of the Spirit. Dario Antiseri and Giovanni Reale emphasize that Hegel´s ideas are impregnate with Christian ideology.

    I don’t know if this is a correct interpretation of Hegel’s theory, but is the one I had in mind when I make that asseveration. I should have quoted them, that’s my bad.

    [1] Apparently this work has only been translated into Spanish, Portuguese and Russian. Il pensiero occidentale dalle origini ad oggi http://www.amazon.com/Historia-pensamiento-filosofico-cientifico-Spanish/dp/8425415926/ref=sr_1_7?s=books&ie=UTF8&qid=1417049752&sr=1-7

    By the way, I think this is my fourth comment, Does the five comment rule also applies to the author of the text?

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  39. @ Patrice Ayme

    “Even if “lay” people and trained critics uttered one million words per second, in the end, all they would have said, and will say in the next century is finite. So I don’t see what the infinite has to do with the subject at hand”.

    Yeah, you are right, independently of how we conceive infinity, regarding to literary interpretation it is clear that it can only be infinite in a lax sense. Because it is impossible to actually known how many interpretations of a literary work can exist. Although arguably if a literary interpretation depends at least in part on the historical context of the interpreter, there surly are at least many, many, many, many, many, many, many, many, many… ways. Just imagine all possible historical contexts, and multiply them by the number of people living in that historical context. Much more possibilities than I can imagine.

    By the way, could we say that “invaluable” things such as art and sentiments are infinite?

    On other topic, I used to think that Badiou had studied mathematics, but apparently his father was a mathematician and he got “trained” in mathematics. I don’t know to witch state he was “trained”. But he is famous for criticizing the absence of mathematics in many of today’s curriculums in philosophy schools around the word. Although I don’t how true this generalization is, at least in the National University of Mexico mathematics aren’t in the curriculum. I won’t be Badiou’s appologist, personally I really like his ideas about the conditions for philosophy, although I find other of his ideas really odd (like his ideas about love… too Lakanian for me). This reminds me of

    @Mario Roy

    “I’m doubtful on this point. Plato’s idealistic and mathematical theory was criticised by the Peripatos with physic and logic reasons. The Cynics issued an existential philosophy in which mathematical reasoning wasn’t relevant, and the Stoa tried an ontological inquiry that matched whit logic or at least was compatible. Parmenides didn’t bother about mathematics to show the Self, and his theory has survived until now”.

    It is so hard to make generalizations in the history of philosophy. So yeah, I agree with you. On the other hand –and I don’t know if the following Badiou’s statement its true- he says that before the “Schism of Romanticism” all philosophers had to be trained in mathematics and science; but since the Schism philosophers aren’t necessarily versed in neither. So maybe that’s what he is referring to when he talks about the divorce between philosophy and mathematics.

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  40. According to French philosopher Paul Recoeur [4], most of the interpretations given to a novel by ordinary people are just quick guesses based in conjectures. For sure, they are valid, but clearly are not as valuable as that of the literary critic, who spends most of her time studying the history of literature and its relation with the social contexts in which it is produced and received.

    I am not sure that there could be any identifiable entity that could strictly be called the interpretation of a novel.

    The Australian writer David Dale once came across a school comprehension test based on one of his own articles. For fun he took the test and found, to his surprise, that he failed it.

    This was particularly surprising since most of the questions were of the sort “What did the author mean by…” , “What is the author’s attitude to ..?” or “What is the author’s tone” sort of questions.

    He contacted the relevant educational authorities and they told him that the questions were not necessarily based on what the author meant or would have considered his attitude, but to what a “reasonable person” would have thought that these would be.

    So perhaps an authors own interpretation of his/her novel would not be considered as valuable as a literary critic’s interpretation of it.

    But if we take an interpretation as meaning the text representation that a particular person at a particular time would regard as the best representation of their interpretation then surely this would be finite.

    Also, I question that the interpretation of a literary critic could be considered the more valuable interpretation. For a start, most novels are not written for literary critics but for the reader at large and so the reader at large is more likely to have to reaction to a work that the author would have intended to convey.

    Moreover, a literary critic is more likely to have an agenda, to be putting more of themselves into the interpretation than the general reader would.

    I recall a literary critic wrote an article on the class politics of “The Lord of the Rings” and the Harry Potter series. His verdict was that they represented an elitist view of class. His argument was that the hobbits represented the ordinary person and that, while they might be brave here and there, the really heroic acts were performed by conventional warriors and kings.

    Anyone who knows the book will realise that this particular literary critic’s interpretation was made without even reading it.

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  41. Coel: “The standard Big Bang model … starts with an infinite extent of stuff around about the Planck time. Thus, … our “real” universe is indeed infinite in extent.”

    The starting point of infinite density and temperature is just a ‘saying’. All equations ‘after’ that saying state a ‘finitude’, all the way to now.

    Coel: “Multiverse models … starting with a finite extent, … where the ensemble would again be taken as of infinite extent. Thus … to one bubble of a multiverse then you’re right that it is finite, but beyond that is more stuff.”

    This is wrong. “More stuff” and ‘ensembles’ will not transform ‘finite’ to infinity in any ‘finite time’. It becomes infinity only if one of the ensemble is infinite or the ‘time’ goes infinite.

    The key issue is how to transform infinity to finite (unknown to the current Math). From the issue of ‘hypothesis of the continuum (HC)’;
    One, Cantor discovered ‘two’ types of infinities, with set theory.
    Two, Gödel/Cohen analyzed HC with ‘consistency’ theorems.
    Three, to show that HC is ‘needed (cannot go without)’ in this universe.

    Again, physics-ism is a triplet (see previous comment).
    P1 = physical universe
    P2 = bio-attributes (not functions)
    P3 = Math

    Then, there are some ‘criteria’ for this triplet.

    Criterion 1 (C1), between (P1, P3): the infinities must be transformed into finites.
    Criterion 2 (C2), between (P1, P2): there must be ‘seeds’ in P1 for bio-attributes of P2.

    The entire mystery of ‘this’ universe is just about these two criteria.

    I have showed that;
    Spirituality = {timelessness, immutability}
    The ‘expressions’ of nothingness = spirituality
    The ‘immutable-process’ > string unification (see, https://scientiasalon.wordpress.com/2014/11/10/the-ongoing-evolution-of-evolutionary-theory/comment-page-1/#comment-9581 ). That is, I have discussed C2 partially.

    Now, I would like to talk about the C1, transforming infinities into finites. This is discussed in the book “Linguistics Manifesto (ISBN 978-3-8383-9722-1)”; thus, I will not repeat it directly here but taking a different pathway, the ‘thread (consciousness)’ which connects the triplet.

    Consciousness is the ability of a self to distinguish itself from all others. The requirement for this is that all entities must be uniquely identified (tagged).

    First, how many entities there are?
    Second, how to tag them?

    For P1, there are at most countable entities.
    For P3, there are uncountable entities. Yet, the majority of the P3 entities are unreachable (by ‘many’ means), see https://scientiasalon.wordpress.com/2014/08/21/defending-scientism-mathematics-is-a-part-of-science/comment-page-4/#comment-6532 . Thus, the key point here becomes for finding out the way to reach (uniquely tag) those unreachable.

    I have showed that all ball-like entities can be uniquely tagged with 4-codes, and ‘all’ entities (of any kind) can be uniquely tagged with 7-codes (see https://scientiasalon.wordpress.com/2014/07/24/clarifying-sam-harriss-clarification/comment-page-1/#comment-5122 ).

    So, if we can show;
    P1 = {7 codes}
    P2 = {7 codes}
    P3 = {7 codes}
    Then, the triplet unification is complete, and the HC issue could be resolved in the process.

    We now at least have a direction to go.

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  42. Coel: I am not sure that is intelligible, nor does it accord with the simple fact that physics accounts for all mass as a collective finite quantity, and not an infinite quantity. The problem as I see it is definitional. If you can reconcile the fairly obvious, unexciting, but likely correct use of “finite” and infinite” as I have stated it (finite quantity of mass) with the idea that infinity exists “at the outset” (from literally nothing in some theories) then I would be happy to read it. But that may not be possible without polluting and misusing a very simply idea – finite versus infinite. The universe you describe using no limit to the “upper bound” does in fact expand. Something infinite in itself doesn’t expand, it is just infinite, but if you use infinite as an adjective, infinitely, then you can apply it to expansion infinitely, which is what the universe may be doing. I realize Coel, and Despicable too, may wish to make more of the idea, to say something just pops into existence from nothing as an infinite state in itself, and yet expands in fact from tiny to huge and continues to do so somehow as an infinite state. Actually it would be better to use a finite state (with finite mass) expanding infinitely.

    You cannot rely on the “unknown” upper bound to say that there is none (as pointed out by Liam). That’s just pulling yourself up by your own bootstraps to somehow get around the misuse of terms in the first place. Inflation is built into Relativity, and that was a rapid expansion. As I say, this is not difficult, or exciting, or wonderfully novel, but nor is it a mish mash of confusion wonderful. It is basic use of language and logic. It might be nicer to have something wonderfully abstract to talk about, but I find it’s better to work with what is intelligible. By the way, if you have an intelligible explanation of a Holographic Expanding Universe under General Relativity (without ill-fitting definitions) I would be glad to access, as would many who just scratch their heads at living in two curved dimensions experienced as three. The other side of the coin in favour of Relativity is that it is local. To be relative is a view from the subjective and entirely local, so it should stay that way and avoid abstractions that try to avoid local confinement by using a Holographic expansion – just more abstracts. If there are no better ways to explain the findings of physics (actual measurement), then find them, I say.

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  43. Disagreeable, I’m going to have to waste a post to apologize for calling you Despicable – in reference to the film Despicable Me, I must have been on autopilot. And double apologies to Liam for calling him Lim, and for rather confusedly mixing up my objection to Disagreeable with my agreement with Liam (no limit to upper bound is not “proof”) in one paragraph above. This thread looks like it took as right angled turn at literature at some point, so no further comments from me at this stage. Sorry folks.

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  44. Hi John Smith,

    Coel: I am not sure that is intelligible, nor does it accord with the simple fact that physics accounts for all mass as a collective finite quantity, and not an infinite quantity.

    I’m not sure where you get the idea that physics treats the mass of the universe as finite, but it isn’t so. As above, the standard cosmological model has the spatial extent of the universe extending to infinity (though see my previous comment for what a physicist means by that). Thus the amount of mass is also infinite. Of course the amount that could in principle be *seen* is finite, given the finite speed of light and the finite time it has had to travel.

    Also, to be clear, in the standard Big Bang model the universe always has been infinite (that is, it starts with an infinite extent, and thus mass, around about the Planck time).

    tienzengong,

    The starting point of infinite density and temperature is just a ‘saying’. All equations ‘after’ that saying state a ‘finitude’, all the way to now.

    The starting point of a standard cosmological model, round about the Planck time, is *not* infinite in density and temperature. The density and temperature were then finite (at 10^96 and 10^32 in SI units). But the *spatial* *extent* was infinite (in the standard model, and again see previous comment for what a physicist means by that).

    This is wrong. “More stuff” and ‘ensembles’ will not transform ‘finite’ to infinity in any ‘finite time’. It becomes infinity only if one of the ensemble is infinite or the ‘time’ goes infinite.

    Not true. If you start with an ensemble of an infinite number of finite entities, then the ensemble is infinite. E.g., if I have an infinite number of 1-meter rules, laid end to end, then the ensemble is infinite in extent. In the same way, both the standard Big Bang model and the multiverse variants start with an infinite extent of stuff.

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  45. Jorge Alex Laris Pardo,
    Well, it seems that Dario Antiseri and Giovanni Reale are saying something similar to what I wrote, only in slightly different terms, and giving more historical context.

    It should be noted that Hegel is not talking about the nature we bump into, so to speak, except insofar as we try to know it – the activity he writes about has to do with phenomenon (in the Kantian sense) and is therefore always activity of the mind.

    How closely aligned Hegel’s thought is with Christian theology has been a matter of debate ever since he came to public attention; Hegel always insisted he was a traditional Lutheran, but his writings look almost nothing like traditional theology.

    Let me here correct the dead link in my previous post: http://www.hegel.net/carlson/Carlson2001-Hegels%20Theory%20of%20Quantity.pdf.

    Finally, I haven’t responded directly to the article, because I am not one of those who thinks that a literary text can have an infinity of interpretations. Umberto Eco pretty well quashed that in his debate with Rorty and Cullers (see his “Interpretation and Overinterpretation”). At some point enough of an account of the text’s origin, reading, and contextual implicature has to be given for the text to have any meaning at all. Successful variant readings then become mere shading (successful in that they don’t dissolve into pure nonsense).

    This is not to say what you have suggested isn’t useful. There are indeed analogous thought processes that can be seen in literature and its study, and in math and physics; as long as we bear in mind that these are loose heuristic tools rather than metaphysical similitudes.

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  46. Happy Thanksgiving to all.

    Coel: “But the *spatial* *extent* was infinite. … both … the multiverse variants start with an infinite extent of stuff.”

    John Smith: “… using no limit to the “upper bound” does in fact expand. Something infinite in itself doesn’t expand, …”

    Thanks Coel. ‘Extent’ is not a precisely defined physics term, but I will not disagree with you on this. Yet, I do like Smith’s view, as the essence of the ‘infinite’ is immutable although the ‘countable’ is viewed as the ‘endpoint’ of an ensemble.

    ‘Infinities’ are the keys for this universe. In my previous two comments, I listed two tasks.
    Task 1, to show that the physics is a triplet and to show the unification among them.
    Task 2, to show that ‘hypothesis of the continuum (HC)’ is needed in the above unification.

    The tactics for the task 1 is to show that all 3-Ps are having the same ‘structure (same dimensions)’ although they have different expressions (body shapes), with ‘consciousness’ as the cut in point. The ‘necessary (not sufficient)’ condition for consciousness is uniquely tagging all entities (whatever they are). I showed (without giving details) that all entities can be uniquely tagged with 7-codes. So,

    For P2 (bio-lives), it is totally described with 7-codes {A, G, T, C, M (male), F (female), K (kids)}

    For P1 (physical universe), it is totally described with 7 codes {R (red), Y (yellow), B (blue), W (white), G1, G2, G3}. There are two ‘color systems’ {quark colors and generation colors} with the W (white) linking them two. The consequence of this 7-code is that Neff = 3.0 [not 2.9 nor 3.1]. That is, no sterile neutrino, no SUSY, nor dark matter particles.

    For P3 (math), it is totally described with {zero (0), finite numbers, infinities}. Yet, is this {zero (0), finite numbers, infinities} a 7-code? This will be the key issue. Fortunately, we do know two things about math.
    First, with 1 (one) and ‘addition operation’, we can generate all finite numbers.
    Second, 0 (zero) = 1/infinity

    Yet, there are two infinity; that is, there should be two 0 (zeros). Again, it is not too difficult to see that there are three types of numbers.

    Type 1, reachable ‘rationally’: 0.333333…333… = 1/3, (type C)
    Type 2, reachable ‘algebraically’: 1.4142135 … = 2^(1/2), (type P)
    Type 3, reachable ‘geometrically’: π = circumference/2 R, (R = 1), (type +)

    That is, there are in fact three ‘zeros’ {type c (0c), type p (0p), type + (0+)}, such as,
    1/3 = 0.333…+0c
    (π) = 3.14159…+ (0+)

    As there are three zeros, there must be three infinities. So,
    P3 (math) also has 7-codes only {0c, 0p, 0+, 1, infc (countable), infp (pseudouncountable), inf+ (uncountable)}.

    This 7-code math is the key issue in the book “Super Unified Theory (copyright TX 1-323-231), and it is available at,
    http://inspirehep.net/search?p=find+a+gong,+jeh+tween
    http://www.worldcat.org/title/super-unified-theory-the-foundations-of-science/oclc/11223955&referer=brief_results

    Now, 3Ps are unified and HC (infp) is found.

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  47. Coel :As above, the standard cosmological model has the spatial extent of the universe extending to infinity (though see my previous comment for what a physicist means by that). Thus the amount of mass is also infinite.

    You are right that no one knows a specific quantity, for example, the number of atoms in the “observable universe” is estimated to be 10^80, and that universe “expands” from an event (Big Bang). It follows that unless you have continuous creation at the Big Bang, ongoing, then it is merely expanding “infinitely” and it is not “infinite”. A better way of saying it is that observation of a finite expanding mass of unknown extent, which is not continually created, inevitably suggests a finite mass.

    You cannot simply jump from a finite “observable” universe expanding infinitely – to infinite mass! Its just an illogical use of terms. Instead say it is finite mass, not continually created, and not doing anything except expand since the Big Bang. As Liam said, you cannot just simply “generalise” from lack of knowledge about what is beyond the “bound”. That bound was once supposed tightly bound as a singularity in Relativity, but so what? that doesn’t mean mass is infinite. In fact it just expands.

    Your terms are skewed as far as I can make out, and likewise your preferred cosmological theory, Relativity breaks down not only logically, but also mechanically in proposing a singularity, and in some variations, it appearance from literally nothing. If you accept that, that’s your choice. I reckon its almost like a con job, the way definitions are skewed in cosmology, to make it more “interesting” perhaps..

    The better approach is to say Relativity is speculation beyond the bound and nothing more. Infinite mass simply means continuous creation, and that theory die with the DoDos. Too many contradictory definitions, but you are right it that the quantity cannot be stated unless the entire universe is observed, but that doesn’t change the logic that it must be finite.

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  48. @ Robin Herbert

    “The Australian writer David Dale once came across a school comprehension test based on one of his own articles. For fun he took the test and found, to his surprise, that he failed it.
    This was particularly surprising since most of the questions were of the sort “What did the author mean by…” , “What is the author’s attitude to ..?” or “What is the author’s tone” sort of questions.
    He contacted the relevant educational authorities and they told him that the questions were not necessarily based on what the author meant or would have considered his attitude, but to what a “reasonable person” would have thought that these would be”.

    That’s a funny story, and it’s quite interesting too. W.K. Wimsatt and Monroe Beardsley had named this phenomenon of seeking for the author’s intention in literature “the intentional fallacy” [1]. Other philosophers of literature such as Paul Recoeur have said that it is not so important to know what the author of a literary work meant to say. Jonathan Culler argued that it’s precisely this characteristic what distinguish a literary text from other types of writing.

    “If in ordinary conversation we often treat the meaning of an utterance as what the utterer intends, it is because we are more interested in what the speaker is thinking at that moment than in his or her words, but literary works are valued for the particular structures of words that they have put into circulation. Restricting the meaning of a work to what an author might have intended remains a possible critical strategy, but usually these days such meaning is tied not to an inner intention but to analysis of the author’s personal or historical circumstances: what sort of act was this author performing, given the situation of the moment? This strategy denigrates later responses to the work, suggesting that the work answers the concerns of its moment of creation and only accidentally the concerns of subsequent readers” [2].

    My teacher of “Hermeneutics of the literary discourse” used to say that the way in witch literature is thought in high-schools -by teaching that the whole point of literature is to “discover” the author’s intentions- should be banned. She was a little extremist.

    @ejwinner

    Thank you for the interesting information about Hegel.

    By the way, it is interesting to note that Culler’s responds to Umberto is based entirely in intersubjectiveness.

    Critics who defend the notion that intention determines meaning seem to fear that if we deny this, we place readers above authors and decree that ‘anything goes’ in interpretation. But if you come up with an interpretation, you have to persuade others of its pertinence, or else it will be dismissed. No one claims that “anything goes” [2].

    Nevertheless, you’ve convinced to add Umberto Eco to my to-read-soon list.

    [1] The Intentional Fallacy http://faculty.smu.edu/nschwart/seminar/fallacy.htm
    [2] Literary Theory: http://www.amazon.com/Literary-Theory-Very-Short-Introduction/dp/0199691347

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  49. Hi John Smith,

    You cannot simply jump from a finite “observable” universe expanding infinitely – to infinite mass!

    Your central confusion is that you are not distinguishing between “the universe” and “the observable universe” — the latter being that portion of the universe that we can in principle see.

    We can only see that region of the universe owing to the finite light-travel speed and the finite time light has had to travel. But this is *not* any sort of edge to the universe! The standard cosmological model has the universe continuing exactly the same beyond that (indeed, it is taken as extending to infinity, though see my second comment for what physicists mean by that).

    It is not sensible to equate the *observable* universe with the universe. For one thing, the “observable horizon” is just a statement about light travelling, not about what stuff exists. Second, the “observable horizon” depends on where you’re looking from. Some stuff beyond *our* observable horizon would not be so if you looked from a nearby star.

    Third, the location of the “observable horizon” changes with time (obviously, as the light has had longer to travel), and thus stuff can move into the “observable universe” or out of the “observable universe” — since the rate of expansion of the universe is not the same as the light-travel speed.

    For all these reasons, it is not at all sensible to equate what we can see (the “observable” universe) with what exists (the “universe”), and no cosmological model does that. (It would be a bonkers cosmology that said that what exists depends on whether it can be seen by one particular species of animal on one particular planet in one particular galaxy at one particular time; cosmologists ditched anthropocentrism around the time of Copernicus.)

    Thus, as in my previous comments, in the standard cosmological model the universe is taken to have an infinite spatial extent (and thus an infinite mass) and always was so, starting off as infinite in spatial extent at around the Planck time.

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