An essay to make *non-creative critters* scream: how do I dare contradict three millennia of revered thinkers and blatant evidence? How do I dare stand centuries of mathematical common sense on its head? Should I be stripped of all my degrees in post graduate mathematics? However, indeed, I dare to observe that: **If It’s Physically Impossible, It’s Impossible: THE INTEGERS USED SO FAR ARE INCONSISTENT WITH THE ( KNOWN) UNIVERSE**.

Some will scoff and point out that mathematics is not constrained by the universe. Really? How absurd can one get? Is not mathematics produced by biological systems? How logical is bio, life? Well, quite a bit, but what life has built, life can dismiss. I am not sorry to offend those who believe it is rational to view themselves outside of the universe, looking in…

***

Abstract: The senior, extremely experienced, and justly famous Princeton mathematics professor, Edward Nelson (with whom I engaged mathematically at Princeton University; he was kind), tried to prove that arithmetic was inconsistent. That would have made him the most glorious mathematician. But he assumed something while deriving his attempted proof, which was not true (as a result his proof was not true either…).

I have more basic, and thus much more drastic, and harder to refute, claims. I play the game… after throwing out of the window one of the most basic “proofs” of mathematics, considered obvious for 24 centuries… I deserve the anti-Fields medal…):

**There is a largest number. Or more exactly, numbers can’t be too large (in sheer size of the numbers of digits needed to express them). All and any logic is bounded, and local.** Full real logic involves qubits, not bits. Only thus is infinity recovered, through non local methods. A situation with realistic logic exists, which closely parallel that encountered in geometry, before the invention of local differential geometry. Local logic can be integrated, using a connection.

In other words, if you can’t build them, don’t pretend unobservable castles in the air exist, and compute with them, to boot! Basic number theory and logic have to become much more subtle.

Specialists may chuckle that all what I am saying boils down to advocating a weaker axiomatics than (say) the overwhelmingly used Peano Arithmetics, and that’s no big deal, just impotent. On the latter point I beg to differ: a stronger axiomatics can confine mathematics. For example Euclidean geometry, which has a stronger axiomatics than Non-Euclidean geometry, prevents to understand highly significant phenomena (like, say, GPS, which uses Riemannian geometry to account for time drift in a gravitationally curved spacetime). Euclidean geometry, though is immensely complex: many axiomatic systems described it, and they are inequivalent. The simplest system of Euclidean geometry is complete and decidable (Tarski, 1948). However Tarski’s theory lacks the expressive power needed to interpret Robinson arithmetic, itself a fragment of the (usual) Peano’s arithmetic. Tarski geometry avoid Godel Incompleteness by being too simple for computations…

Hilbert had desired to found Euclidean geometry upon Peano Arithmetic, which would make it infinite dimensional in its axiomatic structure, incomplete and undecidable (a consequence of Godel incompleteness). So one could make that sort of Euclidean geometry until the end of times. Should we? No. Ultimately , **mathematics should be guided by overall significance.** Doing some type of mathematics suggested by some types of (infinity) axioms, is actually confining minds to focus on dirt, in infinite quasi-loops, ignoring the heavens…

***

THERE IS A LARGEST NUMBER. AND LOGIC IS LOCAL.

A well known theorem in primary school is that there is an infinity of numbers. Indeed, suppose there is not, and N is the largest number. Then the number (N+ 1) is even larger, Quod Erat Demonstrandum.

Simple. That’s what all mathematicians say. But is the reasoning truly valid? Indeed, what is N?

In Cantor’s theory of cardinals, N is the set of sets which have, well, N elements. This is not exactly as circular as it sounds. As John Von Neumann pointed out, one can build up a set with no element (by decree: we just say there is such a thing; it’s an axiom, the axiom of the empty set.) Symbolize it by 0.

Then we can consider the set whose only element is the empty set: symbolize it by {0}. So when you look inside, inside the brackets, all you see is 0, the empty set. Call that set “one”, or 1.Then look at the set having as elements only 0 and 1. One can symbolize it by {0, 1}, that is: {0, {0}}. Call it 2. And so forth.

N+1 would be the set having as elements 0 and N: {0, N}. This way we get all the numbers and the successor operation, +1. So far, this is standard fare, known to all research mathematicians.

However, suppose G* is the apparent number of particles, virtual or not, in the known universe (using the Planck Length which terminates renormalization, and bounds on energy density coming from bounds on gravitational curvature, one can estimate G*; G* is not infinite because the knowable universe is bounded, be it only because, far away enough, space recedes beyond light speed). Contemporary logic and mathematics have ignored this situation, just like Euclid ignored the fact that he did not have a non local definition of a straight line (although he needed it).

Now in the preceding construction of G*, written only as a symbol made of 0s, and the brackets {s and }s, one gets, on the right hand side of G*, well, a large number of symbols }s, namely G* of them: G* }s! That means **one would have as many brackets }s than there are particles, virtual or not, in the universe. But what are the }s made of? Particles, virtual or not. **

**So just thinking of G* is impossible: G* would require all the particles in the universe to symbolize it. **

Some will say: hey, wait a minute, you are confusing mathematics and engineering. In mathematics one generally prove that a would be mathematical object, BAD, does not exist by arriving at a contradiction. Given a set of axioms, AXIOM, supposing the existence of that object, a supplementary axiom, gets to a proposition A such that: A –> Non A.

In other words, honorable mathematical proofs consists in demonstrating that the theory made of AXIOM + BAD is “*inconsistent*“.

Another thing mathematicians do a lot of, as Terry Tao just did to professor Nelson, *who was his logic professor at Princeton*, is to show that a proposed reasoning does not work, because something which was supposed to effect that reasoning, and was viewed as obvious, is not obvious, or is even wrong.

Tao seemed to have found that a sub theory had got to have had a greater *Komolgoroff complexity* than Nelson had supposed; by an enumeration argument. Nelson’ perfect answer: “*You [Terry] are quite right, and my original response was wrong. Thank you for spotting my error. I withdraw my claim *[That Peano Arithmetic is inconsistent].”

My main reasoning here to establish the existence of the largest number G*, is the ultimate enumeration argument. One cannot construct (G* +1) because one has run out of … matter.

Some will say: ah, but to prove mathematics, one uses only the inner experience, whereas you used a mixed approach. Well mathematicians do the same. Euclid famously supposed a number of hidden hypotheses besides his axioms. For example that two circles intersected. The only way to justify that is through *Analytic Geometry* (established in the 17th Century) resting on the concept of continuum (19th Century)… In other words, on the construction of the real numbers, in the second half of the nineteenth century, itself resting on the conventional (and as we saw, erroneous) construction of the integers.

To hammer the point some more. Princeton’s Wiles proved Fermat’s Last Theorem by using some powerful hypotheses about infinity. It is supposed to be a heroic task beyond human achievement to convert the proof into first order logic… And, in any case, it is not clear what axiomatics Wiles really used (did he use an “*inaccessible cardinal”*, in a vital way, or not?) However, as long as the axiomatics is not clear, one cannot assert one has a proof, but just the sketch of one.

Notice that **the main strategy in philosophy, over the millennia, is to precisely show that a time honored reasoning does not work, because something viewed as obvious is not actually obvious, or that is actually completely wrong**.** It’s naturally one of the main ways a philosophical attitude by civilization class scientists impacts science. **

But here we have done something more radical. We have a *symbol which cannot possibly exist*. **No axiomatics can build it.** How could something one cannot even symbolize exist in mathematics?

The limitations on logical systems are also severe and go beyond simply being limited to coding with a finite number of symbols or numbers. **The length of the implication chains** and the length of the descriptions of the propositions, themselves or the numbers describing them are all bounded. (So all diagonalization arguments a la Cantor, including all Gödel theorems fail, etc.)

Thus any logical language is limited, there is a limit to any (local) logical universe.

We will call that the *Logical Horizon*, or *Golo Horizon* (Golo being the male dominant baboon in West African language; there is only that much that a Golo can understand, due to the nature of his neurological universe; also the nickname of somebody dear to me).

**The situation with the Logical Horizon is analogous to the horizon in a differentiable manifold given by the exponential map. Except here it applies to logic itself. Conclusion: arithmetic, and logic are both local. **

(This will have consequences to all domains of thought which use mathematics either technically, or as a source of models or inspiration; that includes philosophy.)

So what happens to the various notions of infinity found in logic? Well, they will have to be reconsidered carefully.

Another notion which can wiped out, is that information is more important than matter: Wheeler famously said at some point that he wanted to reduce physics to information. Or, as he put it, “*it from bit*“.

This is a bad joke if there ever was one. Wheeler knew plenty of Quantum Physics (he was Feynman’s teacher, and co-conspirator at Princeton, after all). Plenty enough to know his joke was deeply misleading. I am myself often reduced to dubious jokes of kindergarten level such as Bushama, Obabla. “*It from bit*” is much worse. Whereas the Bush-Obama era is a solid evidence of reducing taxes on the superrich, giving public money to banksters, warring in Afghanistan, throwing away the constitution, and civilization as obsolete, while describing the whole thing as the opposite of what it is, there is no evidence whatsover for “*It From Bit*“.

All the evidence there is, consists in people thinking that “*digital”* is superior to “*analogue*“. True, monkeys have digits, and they are superior, but that’s roughly where the analogy, and the fun, stops.

“*It from Bit*” is exactly the erroneous conclusion to draw out of Quantum Physics. “*Bit*” is an artificial idea. The real world does not have “bits”, anymore than it has “digits”. As we just saw, numbers are very limited. This means that any physical theory, even a classical one, is indeterminate, just from that.

Any “*bit*“, the smallest piece of information, is a convened packet of energy. In its smallest form it is the presence, or absence, of a photon, neutrino or electron. So any information stream is actually an energy stream. There is a finite number of bits. Fundamentally, because they are about particles, namely, in my vision of the Quantum, very special manifestation of the continuous Quantum reality.

**Reality is all about Quantum Physics, which deals in “ qubits“, not bits**.

**Qubits entangle with each other, are non local, and provide with an infinity beyond integers.**These three complexities that

*qubits*have,

*simple bits*are deprived of. And of course three complexities to be essential ingredients in non local logic.

Information is made of energy and energy is bounded, locally and to infinity, and so are mathematics and logic.

Dedekind famously entitled his work on numbers:”*Was sind und was sollen die Zahlen**“. “What are and what ought to be the numbers”.* He made the famous commentary*:”God created the positive integers, and the rest is the work of man.” *Dedekind made “cuts”… A Quantum event (there no classical events (except in an approximate sense).

However, we just saw that the constraints of the real world are so strong that the numbers cannot be whatever. Maybe, as god does not exist, it could not even create the numbers. Or is it that man created the integers, but, since he was not god, could not finish the task?

Or maybe we just found a proof of the inexistence of god? Behind this joke is a serious point: the idea of god contained that of infinity. However, we just saw that infinity cannot be obtained on the cheap, by piling up numbers in one spot.

***

HOW LOGIC WILL BECOME LOCAL: THE GEOMETRICAL ANALOGY.

The situation as it is in logic, and as I expect it to evolve, is similar to what happened with Euclid. Euclid stricly made geometry on an infinite flat plane, something which obviously did not exist in his world. Or in any world at all. Similarly we just saw that conventional logic and arithmetic do not exist in any world at all. However, qubits are non local, entangled. That allows us to do the same with logic (demonstration some other time).

Let’s go back to the genesis of full geometry. Let’s suppose Euclid honestly tried to draw straight lines on a sphere. Suppose the Earth was an ideally smooth sphere, and one had a bit of straight line on the ground, Bit(1), and a point X off it. Euclid’s postulates said two strange things.

First that the bit of straight line, B(1) could be extended in a full straight line, L(1). That *seemed* obvious on the plane, but it was NOT obvious on a sphere (so Euclid spoke of easier things).

To do this properly, Greek mathematicians would have needed to first find the essence of the idea of a line. That was to minimize length. Now ancient Greeks had to find out what lines minimized length locally, *on a sphere*. As it turns out those lines are what are called great circles.

To figure those out several notions, several subtleties, to extend the notion of straight line to a sphere, a new style of logic had to be introduced, establishing what is now known as *differential geometry*. This immense field of mighty subtleties started in the first half of the Nineteenth Century, with the work of Gauss, Bolay and Lobachevsky, but fully blossomed only a century later, with the implementation of Riemann’s program for gravitation by many mathematicians (and to which Einstein contributed enthusiastically).

The notion of tangent vector was indispensable: this is the direction V in which Euclid would have pointed, when at point x on that sphere called the Earth. The great circle tangent to V is the intersection of the sphere with the plane in (normal three dimensional) space containing V and the vector from the center of the Earth to x.

This can all be demonstrated in various way, the most modern being that the connection on the sphere is the trace of the (“Levi-Civitta”) connection in normal three dimensional space when it is equipped with the normal basic distance known to the Egyptians (the square root of the sum of the squares of the differences of coordinates).

So poor Euclid, trying to extend his bit of line B(1) into a full line L(1), on the sphere, would have been forced to invent geodesics (but that taxed Euclid’s imagination, so he decided to ignore the obvious fact that the Earth was not flat, just like the obnoxious servants of militarized plutocracy nowadays.)

After discovering that great circles locally minimized distance, our imaginary Euclid, if he had tried to implement his fifth postulate *(“Through a point y there is one and only one line, L(2), which never meets L(1)”*), would have encountered miserable failure. However, the very nature of the *geodesics-as-great-circles* would have made clear why: great circles always intersect.

The ancient Greeks could have found out much of the preceding. Actually *Euclid’s immediate predecessors had introduced the first elements of Non Euclidean geometry, with subtle considerations of various angles in possible triangles*. Euclid’s obsessive development of plane geometry was made at the exclusion of the mathematics of his predecessors. It was a rigorous step forward into backwardness.

Why did Euclid do his flat Euclidean geometry, exclusively? Well, I believe, because of the conquest of the Hellenistic world by fascist plutocratic generals of Alexander the Great, who established dictatorships that would last centuries (and similar successor regimes which lasted millennia). A mood set on intellectuals which made it clear that revolutionary thinking was out. And it stayed pretty much out until the European Middle Ages, when the rise of local effective democracy reconstituted progressively the combative originality of the Greek City-States, prior to the Hellenestic degeneracy (while socialized fascism, friendly to demography, but not to revolutionary thinking, installed itself over Vietnam, China, Korea and Japan).

Euclidean geometry was more fascist than the Non Euclidean sort. After all fascism wants rigid, flat, or, better uninformed, uncritical, unidimensional minds, just obsessed by corporate monetary profits. That is why Tom Friedman publishes best seller after best seller, and editorial after editorial in the New York, while that august publication seemed to wisely decide blocking my comments since the *“Occupy Wall Street”* movement has blossomed. More than 50 comments blocked already, and counting… It was the same in 2003 with the Iraq war…

Euclid’s geometry was a physical impossibility on the ground, and that should have given a hint to Euclid’s contemporaries (instead of having to wait 21 centuries, for the obvious). But they had other worries.

We have a similar situation with numbers now. Logic is bounded, finite, and so are numbers, locally. To reach global implications, we have to connect local logics in a global whole.

We have an advantage on the Greeks, to figure more advanced mathematics (and civilization!): we have the Internet, disseminator of truth! And so far just out of reach of the fascist government, in most places. However, have no illusions: so it was with Athens until the well named Antipater took control, after striking a deal with the plutocrats.

**Real numbers are not real. Really.**

***

Patrice Ayme

***

P/S: Euclid’s contemporaries had other worries than finding better, in the sense of more realistic, mathematics, namely surviving fascist dictatorships prone to killing thinkers (like Demosthenes, a philosopher). But we are going this way. September 2011: it’s OK to kill citizens *“when they are hard to catch”, *or* when they edit an obnoxious website*. See: “*Secret U.S. Memo Made Legal Case to Kill a Citizen“*… (Oct. 8, 2011.)

P/S2: Following Hilbert, I used actually Playfair’s axiom (~ 1795 CE) for the parallels, which is stronger than Euclid’s as it excludes *Elliptic* geometries. Euclid’s original axioms just said that if the interior angles were less than pi, the lines met. Considering such angles allowed Euclid’s predecessors to study Non Euclidean geometry (& implicitly contained the notion of connection!)

P/S3: A lot of logic fails, in some circumstances (sometimes the excluded third; for example G* + G* does not exist, although G* does; we also have P(G*+1)= G*+1, etc.). So we definitively have Non Aristolian Logic. Locally.

P/S4:

Many subtileties in logic rest on what can be called “self defeating objects”. Bertrand

Russel concentrated on the set of all sets, and on what he called the “Berry paradox” (he named after the librarian at Oxford who discovered it, he said, in a feat of honesty). In all these cases an object is built, which comes to contradict itself. It is basically always the same object. Godel incompleteness and Chaitkin numbers are examples. A good way to see what is meant is to try to define the first uninteresting number (once defined, it becomes interesting, hence its definition is self defeating).

The point of view here justifies eschews all these problems, radically.

P/S 5: *Why Did Nobody Think About This Before? *Well, mathematicians do not realize that a lot of math is physics by another name. Physicists have, curiously, the same problem: the fundamental axioms of Quantum Field Theory assume that the preceding is false, *AT THE OUTSET. *Indeed the theory assumes that space is made of an infinity of points (with the power of the “continuum”) and that there is an infinite number of degrees of freedom. of course physicists don’t really know the definition of a particle (!), so it does come to their mind that, whatever the definition of “particle”, there is only a finite number of particles in the accessible universe, and that this has some important consequence(s).

Tags: Cantor, Cardinal, Connections, differential Geometry, Largest Number, Local Logic, Non Aristotelian, Non Euclidean, Ultrafinitism, Von Neumann

October 12, 2011 at 7:22 am |

Thanks for using the time and effort to write something so interesting.

LikeLike

October 14, 2011 at 2:57 pm |

You are conflating apples and imaginary elephants. You are trying to compare the idea of a number with whatever it is such a number might physically represent. In other words, the difference between the largest idea and the largest bowl of ice cream, however similar in representation they may appear, are not the same.

I have an apple in my hand. I have one apple. I have a physical embodiment of the idea of one in my hand. I can hold the idea of a billion apples in my hand but I cannot hold a billion apples in my hand. Therefore, I can calculate the greatest number of apples I can hold in my hand but I cannot calculate the greatest number of apples I can theoretically hold in my imagination.The idea of numbers can extend into infinity because at their fundamental level, they are only ideas. Again, the relationship between the largest physical number and the largest idea of a number is strained and inadequate at best.

Furthermore, there is only one rule (and one exception to that rule) with regard to numbers. All numbers (or the numerical idea of numbers) (except one,) designated X, exist in relationship to two other numbers, a number greater than X, X + N, and a number lesser than X, X – N. Of course, the exception to this rule is the number 0 since it is the only number defined by values greater than itself (for obvious reasons.)

As such, we can thus approximate the smallest and largest theoretical numbers in the totality of existence as the solution (0,X) where X remains undefined by virtue of the greater/lesser rule such an actual number would imply. Again, we are talking about the idea of numbers, not what those numbers may physically correspond to. And that is as it should be: actual space always takes up less space than possible space.

LikeLike

October 14, 2011 at 3:59 pm |

Dear Michael: Thanks for thinking about this. And stepping boldly into this dangerous swamp.

This is not an easy subject, and not much progress has been done in 2,500 years, so all ideas are welcome. 2,500 years: it was that long ago that the liar paradox was invented (it was studied in the Middle Ages, and it has been refurbished again and again in the 20C: Russel, Berry, Godel, Chaitkin; each time, it’s pretty much the same sound repeated by parrots with different plumages)… Progress was achieved in other ways. Say with set theory (Dedeking, Cantor, Borel; you seem to be alluding to Cantor’s Cardinal theory).

What you seem to be objecting to is that I get confused between ideas and objects. You distinguish mightily between them: this is the

standard position of mathematicians: unreconstructed Platonism. I mean, this is the definition of Platonism. Ideas are out there, completely different from anything material, so numbers are out there, completely different. Curious they can talk about them, though.I have many objections to this. I will mention two here:

1) So what are ideas made of? If there is an immaterial world, it’s not made from matter, and thus does non-matter, the immaterial, matter? Of course not;

if it’s immaterial, it cannot matter. It’s a …matter of definition. That’s the ultimate self referential, self defeating, argument (I added a P/S 4 on that to the essay).2)

Integers are actually constructed in modern mathematics, using the most elementary first order logic. It is an inductive construction, & the simplest one. It does not require to accept the Principle of Induction (which is second order logic or requires an axiom schema in first logic formulation). The construction is sketched in my essay. It was due to John Von Neumann, and went further than the Dedekind/Cantor level of understanding. My argument defeats that construction for very large integers. An advantage is that many self defeating arguments disappear. Mathematics makes sense at last…PA

LikeLike

October 14, 2011 at 4:46 pm |

Mathematics on many level reflects human views about God and the universe without science. The concept of infinity is an example. Also the Greeks did not have enough information from physical science to inform their geometry; hence, the issue of parallel lines. The idea works on a piece of paper, but when extending to the infinite, a God sort of concept, there’s trouble. Because humans find it distasteful to indicate that the idea of God is just fantastical and accept it as truth, then the flow of knowledge is incorrectly directed. If mathematics and science show that it’s all finite, then why do we keep plugging God as infinite.

You have more understanding of physics than I. Definitely, mathematics is old and some concepts and proofs are just arbitrary constructs that work within their fantasy and beyond.

Regards,

J Ph. MA math & stats

LikeLike

October 14, 2011 at 4:56 pm |

Dear Jennifer: Good points all! ;-)! Math is, to a great extent, abstracted physics. (See various hidden hypotheses in Euclid, such as lines with more than three points, the idea of the continuum allowing circles to intersect, etc.) Thus physics inform mathematics. New physics, new math. Indeed.

BTW, I do believe that all of math, and actually, thinking itself, is just plain physics. OK, the choice of the adjective “plain” is clearly unfortunate here. Verily, we are talking fully delocalized, completely crazy physics here, even more crazy than all known mathematics ever invented… Or, shall I say, constructed?

PA

LikeLike

November 5, 2011 at 7:41 pm |

[…] which we have faith in (I gave a strong reason to believe the PI actually fails in my essay Largest Number, a modest full upside down of all of mathematics and logics that I apologize profusely […]

LikeLike

November 16, 2011 at 11:49 am |

The information on this site is valuable.

LikeLike

November 18, 2011 at 2:06 pm |

I distinguish between ideas and objects (at least 3D objects) because while the differences may seem miniscule when seen on one level, they are quite fundamental on another. In other words, what they have in common, while undeniable, isn’t nearly as important as what they don’t have in common, what makes each distinct.

It might be easier to think of it as a translation line. I can paint and/or photograph a scene that is as close to a 1:1 translation as possible, but not every painting or photograph is completely correlated to an object outside of itself. Think of the M.C. Escher drawing of stairs coming back to meet themselves perfectly. That idea of stairs does not exist outside of its 2D version, and it certainly isn’t used in any 3D version any human being interacts with.

And that is because, what is never said, is that only those objects that are able to transition fluidly between all dimensions become the basis for ideas in the first place. A unicorn is based on a horse, a false statement is based on the rationalizations that make up a true statement (1+1= 2, 1+2=3 leads to the false statements 1+1=3, 1+2=2.) The stairs you can walk up and down upon become the idea for the stairs you cant. And that distinction (stairs you can’t use) is parmount, it defines mathematics and science.

LikeLike

November 18, 2011 at 6:49 pm |

Dear Michael: I failed to understand your idea(s). It would be nice if you explained a bit more.

I would just say that we do not know what ideas are. I suspect they are the fullest, most intricate geometry there is. BTW, we know, more or less thanks to Riemann, or at it least I made clear, here and there, that

any fundamental interaction can be viewed as an evidence of extraneous dimensions. It can be viewed nearly as a tautology. So3D is quite a bit too simplistic. Further dimensions lay in wait whenever one jumps out of the window, or put the fingers in the electric outlet!Now my idea is that,

because of the finite nature of the local baryonic universe, we run out of symbols to build the integers; Von Neumann gets stuck, the Turing tape ends. end of the story. There is a largest integer. So all infinities are not infinite, they have to be considered a different type of process. it is an extremely simple, thus extremely powerful idea.As this modestly stands all of math since Archimedes’ usage of infinity, on its head, I do not expect an enthusiastic rush to my point of view… I have already suffered the uncivil wrath of one guy who became Fields Medal later: mathematicians do not like their brains poked about too much.

PA

LikeLike

November 19, 2011 at 5:38 am |

Simplicity is the basis for all complexity. It is from the simplest of equations that complex equations derive. Put another way, complexity is really just a simple equation that runs 27 pages. As such, understanding and holding onto simplicity is what allows you to understand complexity. Even if there is a dimension beyond this one, the result of the equation of what 1+1 equals does not change. In the 23rd dimension, which comes after the 22nd, 1+1 still equals 2.

And I shudder at the thought of people who haven’t mastered 3 dimensions wanting to find out if there is a 254th. The higher you go, the thinner the logical air gets. Simple is what you hold onto at the highest levels and dimensions. I stopped my metaphor at the third dimension not because I deny any dimension that may follow but because I am still attempting to examine this dimension in full before I continue. Walking is boring but you can’t sprint, physically, logically or mathematically, everywhere you go and hope to reach your destination.

I wonder about your certainty with regard to your statement “…we run out of symbols to build the integers.” Symbols, even words, are not finite resources like the number of pills left in a bottle. Take language, the english language since that is the one I am most familiar with. With 26 basic symbols, there really are no limits to number of words you can create and/or identify as unique. And that doesn’t even include words that function as two unique words in different settings. A frog can croak (meaning to make a sound emanating from its throat) and a frog can also croak (a slang term meaning to die) and not only can my words be understood, the meaning unique to each statement are not affected by the repetition of key words that would normally negate it as such. In fact, the single word ‘croak’ has become so accepted as having two meanings that, depending on the context, the listener might actually fail to understand what you intended because they don’t know which version to insert.

Mathematics (like all sciences) is, fundamentally, a language, and as such, it is equally compatible with any other language in use. And since any other language expands equally to meet another language, the presence of ‘the largest integer’ would simply be assigned some ordained set of symbols and accepted as such. In fact, the presence of such an integer would create a subsequent ‘condition’ upon which its very nature would have to be understood. Thus is the ‘Aymes’ paradigm, based on the Aymes integer (an integer that cannot be added to further and represnted as A], A for Aymes, bracket for enclosure and the momentum being pushed back upon itself) birthed into being.

In science, as in alot of things, if you build it, they will come and name it.

LikeLike

November 19, 2011 at 6:52 am |

Dear Michael:

Thanks for giving some more thought to this very tough problem. OK, before I get into splitting hair about the question that the more dimension, the more complicated, let me try to understand your main argument, which is actually connected to something very deep, I feel. What you are saying is that, if there was a largest Ayme integer, it could be named (“

assigned some ordained set of symbols“), and then “accepted“. OK.That is a very old question, philosophically. It’s tied up to the problems of universals in Plato, and theories such as nominalism and conceptualism, or realism. it was a raging debate in the Middle Ages. If my reasoning sticks, this is a major twist in the debate.

In math,

main stream conceptualism claims that(MY formulation/interpretation here). Problem: little of math has been reconstituted nominalist style.second order logicis inadmissibleMy own position is complex and new. I am a sort of conceptualist (it’s all in the mind), but also a realist (real structures in the mind). But I am also, thanks to my largest number reasoning, a form of extreme nominalist never seen before, the FINITE nominalist.

So I am a lot of things which are supposed to be incompatible (and I will not pretend that I have it all figured out!)

What I just pointed out is that the usual construction of Peano arithmetic (= adding + 1), even in the simplest Von Neumann style (sets made of sets built from the empty set and brackets), runs out of symbols. Renaming is not an option; the Ayme number is designated by all and every particle in the (local, accessible) universe already.

The Ayme number is the cardinal of the universe actually. So one could designate it by 1*. It seems to be what you suggest. And then we would have all the number until Ayme], and then 1*. Actually standard ordinal theory does something a bit like that. 1 + aleph zero is aleph zero, but aleph zero +1 is not aleph zero…However cardinal Ayme] + 1 does not exist: if it did, a set would be larger than the Ayme number, but no set is larger than Ayme, because the Ayme number contains already all the particles in the universe, just to designate it, whichever medium you use]!Rabid Platonists, still not understanding the point, would say that they decide it should be so. So they put in their brain a set of cardinal Ayme], and then add one element. However, to designate Ayme] they have used the whole universe, so their brain has no particle left. This proves actually that if true Platonist exist, they have no brains. QED.

BTW, in a slightly more serious vein, and just a detail: one should not assume that adding dimensions makes things more complex.

Some topology is more complex in lower dimensions than in higher dimensions. Steve Smale proved the general Poincare’ conjecture for dimensions five an higher long ago, in the 1960s. He got the Fields Medal. Then Mike Friedman proved it for dimension 4 in the 1980s. He got the Fields medal, too.

Then Perelman, building on the work of others, as we all do, proved it for dimension 3 (the original conjecture). He then rejected the Fields medal. And I approve.

More prosaically dimension 2 analysis (so called complex analysis) is simpler than dimension 1 (real analysis), because every polynomial of degree n has n roots…

PA

LikeLike

November 19, 2011 at 12:45 pm |

I think I am beginning to understand where we might be having problems understanding each other. For me, 1+2 directly correlates to the idea of a+b. In other words, I have to be able to put an actual integer into any conceptualized set in order for that set or equation to be valid. That is why I understand a concept such as the square root of a negative integer to be equal to that of a dancing unicorn playing the banjo. A mathematical set that conceptualizes an integer that doesn’t correlate to any known or applicable integer (or the basic rules of integers, where n can always become n+1) is akin to imagining that dancing unicorn playing a banjo. You can build an animatronic unicorn playing a banjo but its correlation to horses and musical aptitude is specious.

I also think we are ignoring where physics and mathematics actually meet. Let us assume I have just entered a room with a chair (a comfy chair because I’m going to be here a while.) I sit down and start counting: 1, 2, 3… The rate at which I am counting is consistent. If we ignore the biology and physics for a moment, we can safely assume I can continue counting forever. However, the numbers I can count are as much tied to when I started counting and the rate I began them as the numbers themselves. If I had started counting sooner, I would have a bigger number than if I had started later. Furthermore, if I increased my rate of counting, I would reach bigger numbers than if I had counted at the same rate or slower, but that would be impinged by the determining factor of when as well. And all of these factors end up influencing the ‘largest’ integer I ‘stop’ at. And that also influences my counting because the fact that I actually ‘started’ counting implies a moment before when I wasn’t, which determines that any number I arrive at will and must be smaller than the potential number I could have arrived at. The potential number MUST always be bigger than the actual number, but the physical properties of existence only allow us to ever know the actual number we encounter.

I use a similar thought when I want to daydream away my problems. I imagine myself exponentially travelling away from Earth in a straight line. I then exponentially compress the amount of time I have been travelling. At some point, I always stop. And when I stop, I always know that I have not actually ‘escaped’ the bounds of Earth becauce the path I have mentally took has created a diameter, which allows a radius to be created, which forms a circle (both 2D and 3D) which, in essence, doubles the amount of distance I traveled. Even when I was racing away from Earth as fast as possible, my shadow was racing in the opposite direction, laughing at my efforts the entire time. The largest circle in existence still conforms to the basic mathematics of geometry.

LikeLike

November 19, 2011 at 5:33 pm |

Dear Michael: As a poet, I do not know how much math you had. You seem to rediscover pieces of reasoning long established, like Euclid’s proof that there is NOT a largest integer. That basically says that if N was largest, N + 1 would be even larger, so there is no largest. Fine. My objection is that Euclid’s reasoning fail, at some point, not because (Ayme] + 1) is not larger than Ayme], but because (Ayme] + 1) does NOT exist. The (serious mathematical, state of the art) existence of integers basically assumes that different integers have different markings (something already discovered in the early Neolithic). I just observed we would run out of particles for the different marking, because the local accessible universe is FINITE.

In a way my observation is trivial. It was not made before because scientists and mathematicians are (over-)specialized, they can’t add up disjoint notions, and miss the big picture. Same with my observation that Relativity obviously fail at high energies.

In economy, it’s worse. Money has been privatized, and no civilization with privatized money has left an enduring mark…

PA

LikeLike

November 19, 2011 at 8:04 pm |

Running out of an actual particle and running out of the mathematical idea of particles are two different particles altogether. I can use the word ‘passenger pigeon’ without there actually being a passenger pigeon in current and expanding existence upon which I am speaking. In that regard, the ‘idea’ of the passenger pigeon has outlived the actual species to which it refers. The old question of “if a tree falls in the woods, and no one is around to hear it, does it make a sound” makes the false assumption that trees only make sounds because we have ears. Trees have to be able to make sounds before we can hear them in order for them to make sounds we can hear, which must, consequently, be able to make sounds even when we go ‘deaf’ again.

The only way to assume what you are attempting to assume would be to assume that space is a function of mass to begin with, and that is perhaps why I am having a difficult time accepting your rationality. You assume both the particle and space cease to exist at Point Ayme (which would make your observation true and such a demarcation the ‘largest’ or farthest) instead of thinking, as I do, of space as being undefined and thus able to be infinitely (or at least potentially) expanded into. The assumption of N+1 always assumes some form of ‘space’ (even if its only theoretical) to expand into. But if even theoretical space itself were actually finite, then you would eventually reach Point Ayme and have no physical mathematical or existential choice but to eventually fall backwards.

I always assume a space bigger than my existence. Because of that, even if I go as far as I can physically go, I still can theoretically go farther, if only in my mind. Whether or not that is mathematically or physically supported is another matter altogether.

LikeLike

November 20, 2011 at 4:07 am |

Dear Michael: You make several astute points. Overall you say, it seems to me, that my reasoning claims to run out of integral numbers, because, basically, I claim that I am running out of thought, and, or, space. Well, I agree. I have long pondered what thoughts are, and what space is. well, i think they are all made of arrangements of (the geometry of subquantal) matter.

You seem to hold on the hard Platonic position, that ideas are immaterial. Most mathematicians subscribe to it, but I completely disagree, as I just said. If ideas are not material, then what are they? It’s a bit the problem of the multiverse: one cannot explain something, so one says it’s out of the universe. But that is not to understand the concept of UNI-verse.

And I have a further twist to annoy Plato: if space is not material, then what is it? “Einstein”‘s theory of gravitation has nothing to say about this, nor Quantum Field Theory.

PA

LikeLike

November 20, 2011 at 2:19 pm |

Ok, let me try a different path. I exist in a physical sense. I know that I am, as Descarte would say. But in knowing what I am, I know what I am not. And I know this because of the physical properties that define what I am and what I am not. In that regard, what I know and what I don’t know is still based on the differences between ‘stuff’. An apple is not an orange but both are made of some form of stuff.

But this first dichotomy of differences of stuff leads me to make a different leap. If there is such a thing as two different kinds of stuff, if there is even such a thing as ‘stuff’, is there such a thing as no stuff? Being as I am made of matter, I could never actually interact with this ‘non-stuff’ because, like the equation 1+0=1, my very stuffness would be forced to reflect back upon itself. If I step into an empty room, the only thing I am feeling and knowing is myself. And everything I used to describe or understand this ‘non-stuff’ would of course fail or be inadequate because I am using stuff (words, phrases, sounds, symbols, etc) to describe something that doesnt use words phrases or sounds. If you take the absolute approach that only a non-stuff approach can accurately describe something that is non-stuff, you are correct. But that isn’t the failure of any particular symbol but within the very flaw of ‘stuff’ to begin with. Thus, I use a word like ‘space’ somewhat interchangeably with a concept like 0 because it fits the physical properties of an object (1) occupying space (0) and thus allowing it to exist (1+0=1.) If you wanted to rewrite the equation as it physically occurs, it might look like 1=1+0, with one recognizing itself and the empty ‘space’ which it occupies by giving it an ultimately inadequate symbol the 1 can relate to. 0 therefore becomes an ‘idea,’ a manifestation of the 1, and thus made of stuff, but that representation (again, inadequate at best) allows the 1 ‘space’ to move.

This idea would necessarily contradict Einstein’s version, since he assumes space to have some base physical property (like people swimming towards each other) that can be interacted with. The thing is, friction between objects moving against each other makes them less effecient. Think of how much effort you have to use in order to ‘walk’ in 3ft of water from one side of the pool towards the other. “Empty” space (or space dominated by particles very far apart from each other, thus lowering their friction) would appear to make more physical sense. Less friction means more energy used by the particle to actually move. Between walking in water and walking in ‘air’, assuming no other choice or influence (like something on fire close by) wouldn’t you choose the least resistant, the least dense option?

But don’t confuse my use of the word ‘space’ to think that I think all ideas are immaterial. In fact, I already assume failure when I use a concept like space or no-apple because I understand and accept the contradictions already in place. The idea of no-apple and the idea of apple are ideas emanating from an idea creator, a thing, a conscious pile of stuff. In fact, it might be said the only things I consider immaterial is non-stuff but I still use a material, physical thing (idea, words, symbols) to describe it.

LikeLike

November 21, 2011 at 1:46 am |

Hmmm… Let’s say this: in modern physics, zillion attempts have been made to give some structure to space. Out of these, many different structures are used, and viewed as “true” to some extent. Like the one in Relativity, or the one (those?) in Renormalizable Quantum Field theories. Speaking of feet in water, various versions of the Higgs Field…

Some of these aspects of space are demonstrated, such as the Casimir effect (let alone renormalized effective field theories…)

All this is fine, and here to stay. Just aspects of space. But very far from the final story.

Now, as a logician, there is, as you yourself seem to point out, no difference of nature between proposition “A” and proposition “NON A”. So, a fortiori, if “A” corresponds to something material, as I believe it does, so does “NON A”. Some material-geometrical arrangement in 7 billion brains. 7 billion times, more or less reproduced, sort of the same.

My largest number idea is thus just the statement that, having a finite number of particles, locally in this local accessible universe, we can have only a finite number of ideas.

Thus basically much mathematics is an infinite approximation to a finite theory!

Having thus stood the world on its head, I will now rest, with my baby.

PA

LikeLike

November 21, 2011 at 3:57 am |

I knew I was in trouble when I formulated that idea, but I had to continue in order to make my point. Yes, I agree that the idea of non-A must, by the very rules I laid out, be material and thus have some basis in reality. But its a reality that is compositional in so much as it is made out of parts of matter and reality but make it a reality with a particular designation within it.

It’s kind of like the reality of a lie. To say a lie, to think a lie, to speak a lie are all means by which the reality of this thing called a lie is understood, created, and interacted with. But its reality is tied precisely to what it isn’t, what some ‘true’ reality is providing its foundation for. While 2+2=4 and 2+2=6 are equally real, they are not equally valid. And even if the symbol ‘4’ is arbitrary in its designation as being the result of the product of 2+2, it is no more or less arbitrary than the symbol ‘6’ which means what matters more is not what the symbol is but what the symbol’s meaning is agreed upon.

Now that I think about it, I think I am most bothered by the word ‘can’ in your statement “we can have only a finite number of ideas.” If you rewrote the phrase to say we will only have a finite number of ideas, I would probably be forced to agree with you. And that’s because you could rewrite the phrase to say “we will only have a finite number of heartbeats” or “we will only have a finite number of tacos” and the statement (in so far as it will be true for any said individual) must therefore be true and an aspect of reality. In that regard, the number of tacos, for example, a person can eat (because they are a human being and finite in their existence) must be bound to a specific number, a number that cannot go further. It doesn’t matter if a person eats tacos morning noon and night for the duration of their life, there must be a ‘final’ taco.

But where I have the problem is the result of that final taco (or your explanation of it) seems to undermine the very process that lead to it. For one thing, it ignores how a ‘finite’ number of particles could recycle itself to represent more than a single ‘thing’. When I eat a taco, the taco is broken down into waste, which gets pumped into some sewage system, which gets treated and turned into fertilizer, which gets thrown on the ground to augment the nutrients which will be absorbed into the plant that is harvested and taken to some market, where it is bought and transported to some kitchen where it is made into a ‘new’ taco. But how can it be a ‘new’ taco with an old particle in it? If you think about it, there are only 9 unique numbers, plus zero. Anything bigger and they just perform some function of repeating. I have more fingers and toes than there are truly unique numbers in existence. In that regard, we already know the largest unique number already and its called 9. As such, we thus know what happens when existence reaches such a point: it attempts to uniquely repeat itself. You were looking for a number you knew all along.

LikeLike

November 21, 2011 at 4:30 am |

There is something I just thought up that also makes me wonder about your idea of ‘largest number.’ I bet you could take every number in existence and attach some piece of trivia to it so that it would read something like “X is the largest number of toothpicks made at the factory in Shanghai.” If you think about numbers in that fashion, each number already functions in reality as some largest number addendum already, which is how you began to build towards your largest number idea to begin with. But while your number would be the last number, would it fundamentally be different or so unique from, say, the second-to-last number that you could not call it a number?

LikeLike

November 21, 2011 at 5:09 am |

Dear Michael: What you are alluding to, without being aware of it is the paradox of the Least Interesting Number. But what I am talking about is different. It’s more radical. More at the root.

As Malthus pointed out, there are limits to growth. In a way, that’s all I am pointing out. To point out that there is recycling does not change my labelling argument: one cannot recycle the labelling as it is going on.

PA

LikeLike

November 21, 2011 at 4:55 pm |

After thinking about it some more, I came up with a different idea. You asked what the largest number was in regards to something like particles. But what you are asking, in terms of physics, is the largest amount of particles available in the universe. But that number actually isn’t the largest.

A particle, by its very definition, is matter but its only a piece of matter. It is like taking a pie and cutting it into 8 pieces. The number 8 is bigger or larger than the number 1, but its only referring to pieces of a whole. In other words, there may be 8 pieces of pie but in actuality, each piece is only 1/8th of some larger whole. Demarcating smaller and smaller pieces of pie give you a theoretical larger number but each number is equally tied to its exact smallest counterpart because each represents a piece as much as it represents itself. Therefore, the actual largest number is 1, the total of all pieces taken together to constitute mass as a whole. Every idea is really a demarcation, a piece of the actual largest thing called ‘ideas’ which is still 1 thing.

Malthus’s Catastrophe assumes a planet of 6 Billion people living like Bill Gates. But while that catastrophe might occur if such a thing were possible, we couldn’t actually have a planet composed solely of 6 billion BIll Gates because Bill Gates doesn’t drive his own car or make his own McDonald’s hamburgers. Therefore, lesser consumers have to make up the population of consumers and be taken into account. As such, what we then divide the planet into is people who use vast amounts of resources, in short amounts of time, compared to people who use less amounts of resources over a longer period of time. But that always assumes the resources we currently possess. In other words, 6 billion people lighting their homes based on coal is different than 6 billion people lighting their homes based on solar power. Depending more on resources that are renewable allows more people to depend on them without reaching Malthus’s conclusions while also affirming his conclusions at the same time.

The only way to think of growth as being limiting is akin to thinking of Earth as being a clown car and calculating how many physical bodies can actually fit into the dimensions of Earth. It is not that such a number could ever actually occur (we’d reach Malthus Catastrophe long before that) but the potential number of human beings you could stuff (if you were just counting them as physical spaces) could then be calculated, according to the laws of area and dimension. X amount of area in total divided Y mass per person gives you Z amount of physical people that can inhabit said space. In that regard, there is a finite number of human beings that can exist on Earth but it would only ever be a theoretical number. It would only ever be a potential, never an actual or realized, number. And even that number could change depending on the particular mass you assign to each person. Are we talking 6ft linebackers or 2ft skinny dwarfs?

LikeLike

November 22, 2011 at 11:27 pm |

7 billion, as we have, soon to be 9 billion, with today’s tech, is just untenable…

LikeLike

November 23, 2011 at 4:24 pm |

I agree that 7 billion with today’s tech is untenable. But is that because of the population, the technology, or because of the interaction between the two? A majority of those people are poor or high consumers meaning they use up resources in such a way as to render them untenable. Lets not forget the way in which resources are acquired has as much to do with how tenable they are as who is buying them. Small amounts of people, with regard to the population as a whole, using technology to trawl for fish can degrade the resource even if only a small amount of people are actually the consumers.

Also, culture has as much to do with sustainability as anything. If a culture (such as Japan) bases its identity around sushi, as that population and access to technology grows, even poor members of the community will take part in the resource’s extinction. What % of the world’s population does Japan represent or even what % of seafood consumers does Japan represent? They perfectly represent a small amount of people using vast amount of resources based on cultural identification. Therefore, the sustainability of the oceans has less to do with more people using them than the means by which those increased amounts of people obtain them.

Price has alot to do with resource management as well. If the cost is unrelated to its actual availability (tuna should be higher than it is,) that will further the extent of the degredation because people will not self-regulate. Until the last can of tuna is actually made, 99% of the consumers won’t understand what was happening. And consumer ignorance has as much to do with sustainability as how many consumers there actually are.

Malthus’ Catastrophe does have a finite number but its based on a set of conditions that can change. Furthermore, if it is true that 7 billion can’t be sustained, wouldn’t a population drop (by whatever means) still force the catastrophe to continue? In other words, is the Tipping Point the point at which all life has no choice but to go extinct but the point at which a different set of circumstances must be achieved in order for some form of population to remain? A tainted supply of beef killing a billion people might bring up below Malthus’ point (even if we were previously above it) which gives us more time to get things right. And getting things right is what catastrophes are really all about.

LikeLike

November 25, 2011 at 1:54 am |

Michael: The situation now is ominously similar to pre-1914: then everything was swell, and many factors of globalization were equal, or actually higher than now (could go nearly anywhere without a passport). And then there was the terrible crash of 1914, and a partial recovery was not achieved before ~ 1950.

Now of course in 1914, there was a blatant villain: The German Reich, a mass homicidal fascist regime, playing superpower (it was relativity as large as the present day USA). There is nothing comparable today: whatever China’s defects, it’s not mass homocidal, and it embraced probably too many of the aims of the 1789 revolution to fall into uncontrolled fascist plutocracy.

However, now as then, the tipping point will probably a military one. In 1914, the Prussians attacked, and created a world war deliberately, thinking they could win it. Nothing like that looms today. But, because the weapons are much formidable, a smaller power could still lead everybody into war. If a war ignited between Pakistan-India-China, the West would find impossible to stay out. As it is, first strikes would be tempting.

In any case disruption of the world economic order would lead to massive starvation.

Worse: the world hitting Malthusian limits could lead to war situation which themselves would be quickly amplified by war. An obvious near term trigger would be war with Iran, and the closing of Persian Gulf oil. Completely crazy scenarios are imaginable (such as a hit on the world’s largest oil field with a dirty nuclear device).

In any case, when the crash comes, the death of several billions is a realistic outcome.

PA

LikeLike

November 25, 2011 at 11:00 pm |

You bring up an interesting point about how a Malthus Catastrophe might come about. I was thinking in a strictly all-things-being-equal scenario where finite resources meet unregulated growth. Bringing up war and such raises an interesting point since they may or may not actually be related to Malthus’ idea at all.

The thing is, even the most healthy of systems (biological, economic, social, etc) can be made to collapse if hit hard enough. Would such an act actually qualify as being a Malthus Catastrophe or would it be artificial? In other words, a society optimally regulated and living within the means of its environment could still be wiped out by outside forces that were outside its scope of accountability. You then couldn’t blame the extinction of the society as the result of a Malthus Catastrophe because you’d have to go all-in and say that the world would have to become a single society in order to escape any factor of ‘outsider’ in such a calculation. Or, you’d have to allow that Malthus’ Catastrophe isn’t so much a warning as a reminder, since the very finite and fragile nature of existence would ultimately precipitate some form of destruction that would inevitably lead towards collapse. A society that relies heavily on farming would be devastated, perhaps fatally, with the consequence of a single season of drought but even a multi-facet society would still have pressure points that allow it to become vulnerable to any shock that might occur naturally or artificially. Think of the social and economic impact a 9.8 earthquake hitting Los Angeles would/could have. And that could happen anywhere: plains are prone to tornadoes and violent winds, river communities are prone to flooding, the list of dangerous places eventually makes the entire world unhinabitable for any certain length of time. And those are just the natural forces at work. Take into account the irrational instinct of man and no form of social order is sacred or safe.

And while I do think there is the smallest number of people who need to exist in order to maintain population without massive inbreeding and dying out through mutation, any potential catastrophe (natural or man-made) that could inflict such would imperil the survivors as well, since it could easily go beyond the point of leaving enough viable humans left. But that can happen at any point in history. Even a caveman with a really sharp knife could extinquish a vast portion of early human beings, not leaving enough left to allow the popluation to continue. I’m not quite sure Malthus was talking about sociopaths when he came to his conclusions.

LikeLike

November 25, 2011 at 11:34 pm |

Malthus did not seem to have anticipated a crash, indeed. Nor does Pinker in his latest book on how smart and good we are. The more advanced a society, the worse the crash, of course. Just like if one flies at mach 3, and then crash, it’s worse than if one runs along at 9 kilometers an hour, and then crash…

PA

LikeLike

November 27, 2011 at 4:24 am |

I’m not quite sure about advance technological civilizations being any more or less stable than less advanced. I think its more six-of-one, half-dozen-of-the-other.

The assumption that greater speed always leads to a worse result tends to ignore factors such as availability of medical knowledge compartive to the incident. Despite the number of automobile fatalities, and the speeds at which they travel, safety features mitigate the amount of human carnage that actually does occur. Likewise, in very basic socities, falling into a next of pit vipers, no matter how slow you are going, isn’t going to produce a very pleasant result. Basic cuts and diseases in a ‘slow’ society can be life-threatening. Mother and/or child mortality rates (at least in Western socities) once made getting pregant as fraught with danger as any technological advancement in an advanced society. Flying in an airplane is dangerous but essentially voluntary. However, if the very act of sustaining the society can kill you, how can you equate the potential crash as being less so? I certainly don’t long for the routine pandemic days of yore nor do I look back fondly when ye olde medical advice required the use of leeches to correct the imbalance of humours in the body of a sick patient. If the only way to get an average life expectancy of 75 years is for some people to go Mach 3, and potentially crash, I’m sure alot of people would gladly make that trade. There is a reason why most societies advance despite the dangers.

LikeLike

November 27, 2011 at 4:38 am |

Clearly the prehistoric lifestyle was stable over at least 2 millions years, about 200 times longer than Western (in the sense of Anatolia-Fertile Crescent) civilization. The case of the Maya shows that an advanced civilization with writing and some math can collapse on its own. Clearly, in that case, only more advanced tech, or intellect, would have saved the Maya. Same for the Romans.

It’s also very clear now: our civilization is unsustainable with a global population approaching now ten billions (because that’s what 7 is: not far from ten),

while restricted to present technology. So either 90% of humanity is promptly eliminated, and we achieve sustainability that way, or nuclear energy makes great strides in the next decade or so, and we get to a clean future.At least nobody will say I am not provocative…

Although, with my largest number, I went already further than far out…

PA

LikeLike

November 27, 2011 at 5:30 am |

I’d be careful creating a causality between stability and the prehistoric lifestyle. It could easily have been that with fewer members, a single catastrophe could have set them back to square one any number of times. Assuming two millions years of stability and two million years of repeated cycles of slow growth coupled with abrupt depopulations are two different scenarios. With no writings to pass on, the death of a storyteller/elder could be quite catastrophic in a nascent tribe. If you lose your map, you go down to the local gas station and buy a new one. You lose your tribe’s mapmaker, it’s back to eating beetles and cave moss until you (or what’s left of your tribe) rediscover the fishing area again. And unlike animals, mama dying in the cold pretty much ends the life of any babies present as well.The prehistoric version of “man bites dog” was “child survives without parent.”

But back to current dilemas. No, nuclear power is never and can never be the solution until the problem of radioactive half-life is taken care of. Anything that creates an utterly toxic byproduct that lasts thousands of years is NOT something you want to base your technology around until you neutralize that technological ‘hiccup.’ In some ways, nuclear war is less of a threat than nuclear waste: we can see the immediate effects of a bomb going off, we can’t see or comprehend the long-term effects of storing such waste in sites no one is going to want to pay all that much attention and detail too. How many spent fuel rods does it take to irradiate a groundwater source for how many thousands of years? And I trust the casks they build to house them as much as I trust the warranty on my toaster to cover any problems it might encounter. LG skimps on the warranty, I buy a new toaster. GE (or whoever) skimps on the quality of casks it builds (or simply under-rates them to begin with,) California glows in the dark for more than just because of the lights in Hollywood.

I don’t consider you all that provacative, possibly because I see pushing the theoretical bounds as being necessary to the job. You can’t simply assume, you have to take an idea and push it as far as it will go, sometimes even beyond it. Once you know the point at which something works, and which it breaks, then you know where things need to improve and where things need to be invented whole cloth. The line where the possible and the impossible meet still needs to be known, and updated if necessary, but that won’t happen if nobody ever tries to go past it. Nobody ever ran a marathon by accepting the notion that they could only run less than.

LikeLike

November 27, 2011 at 6:50 am |

I am searching for a concept beyond civilization. Civilization depends upon the notion of cities. Culture, even chimps have. Culturalization?

No doubt zillions of prehistoric cultures, races, and even subspecies were annihilated. We actually now have the proof. But the point was this: the level of knowledge necessary to survive was independent of the environment. It’s not the case anymore.

For nuclear, I am of course an enemy of the present nuclear technology (Fukushima is made of Generation One Power reactors!). The concept of nuclear, though is irresistible:

the energy densities of nuclear fusion are one million times higher than the highest chemical energy densities, and one BILLION times the energy density of the most performing batteries.Nuclear waste does not have, in theory, to occur. It’s an infortunate consequence of the present type of nuclear tech. There was a nuclear leak over Europe a few weeks ago, due to a medical facility. Nuclear waste from medical is a very bad problem: maybe we could let die all those who need radiation? Of course a thermonuclear flame would solve the nuclear fission waste problem.The confusion between waste and nuclear is similar to the confusion between human waste and humanity. Except, for nuclear, waste is not a necessity, but it is in humans. Transmutation has long been demonstrated. That means potentially all and any nuclear “waste” is, potentially, fuel. Thus, there is no problem. Moreover nuclear waste is highly dense, and easy to confine, in striking contrast with CO2.We are at a state of tech where we can develop nuclear, among 100 possible techs. If we do not, the moment will pass, the energy to make the research will not come back, civilization will regress, and war will come… Better a bit more radioactivity now, here and there, than human oil lamps, believe me. The funniest thing being that moderate, very moderate radioactivity improves health (it kills at higher doese, that’s how and why FDA was created).

PA

LikeLike

November 29, 2011 at 4:45 am |

I think knowlege and environment go hand in hand. You can’t tell me surviving at the equator and surviving at the poles require the same skill set. In fact, it would probably have been something like Darwin’s tall and short giraffe’s metaphor. The tribes most able, or lucky, to adapt to their environment thrived while the rest migrated or went extinct. Even if Native Americans were simply the offshoots of some nomadic Russian/Arctic tribe, the imposition of their environment forced them to adapt, especially as the Ice Age ended. Wearing fur in Arizona and wearing fur in the Russian steppes, however related they are, aren’t done for the exact same reason. In that regard, the artificial environment of cities really is like an urban jungle, albeit twisted. In some ways, humans have actually swam against the species current, since many of our celebrated examples aren’t very capable of suviving in any environment outside of their own habitats. Humans actually promote traits that don’t provide a long term benefit, which is something new and dangerous to the environment.

In that sense, developing nuclear power, and waste, before we were capable of responsibily dealing with it might actually be another symptom. The potential for massive amounts of energy from relatively small amounts of fuel is irresistable, but its obvious the science is still lagging the technology. I did always wonder why they couldn’t use the still-charged fuel rods as some sort of slowly decaying battery. Isn’t there some sort of material that will absorb the charged particles along its surface and turn it into electricity? Not taking advantage of the secondary heat that fuel rods give off really is wasting an awful lot of potential energy. And I’m not that current on my nuclear physics so my understanding of thermonuclear flames is sketchy. I can only assume they burn hot enough to eliminate the half life of the material without going full-blown chain-reaction somehow, but that’s all a guess. Again, another scientific/technological idea with loads of ways to be poorly implemented only gives me another reason why nuclear makes a poor fuel to be used as a base for civilization and technological.

Personally, I’m still waiting for a hydrogen engine that produces water vapor or even oxygen as a by-product.The waste could then be stored and used as coolant to bring the total engine as close to 99% efficiency as possible (I doubt any engine or process will ever be 100% efficient, so I’ll aim just a tad lower, although I understand even 99% is still high.) It wouldn’t solve human ignorance or human stupidity but at least we’d all be enslaved in a more environmentally friendly way.

LikeLike

November 29, 2011 at 7:43 pm |

A hydrogen engine where H2 is combined with O2 exothermally cannot be 100% efficient: it’s a Carnot engine, and at its most efficient, maybe just half as efficient of what a fuel cell can deliver.But burning is easy, fuel cells are hard (see Apollo 13: BOOOOM!).

Many nuclear engines can burn waste. The French make MOX (Mixed Oxyde), which basically burns nuclear waste. ALL French reactors work that way. US Congress has made MOX use unlawful. The French converted US nuclear warheads in to MOX, and send it back to the USA, to respect the START nuclear warheads reduction accord between the USA and the USSR (so called Russia).

Fast Breeder Reactors do burn waste and create more. The French Astrid FBR aims, among other things to demonstrate elimination of waste by transmutation. Fusion reactors, with 100 million Celsius would just disintegrate radioactive waste.

An FBR on the Caspian sea (with which France helped) was also used as a desalination plant to furnish the local city with water in the local desert…On going cooperation with Russia and Japan goes on, India has a very extensive program centered on Thorium)

I have enormous contempt for some of the people who use ecology to advance themselves (there are plenty of those in France, in particualr an hysterical activist called Duflo, a manipulative arrivist, who would have been an SS guard in other times and places; Gore is a bit like that too, but, overall his action was more positive than negative…)

Tech is the only true friend ecology has.PA

LikeLike

December 13, 2011 at 9:17 am |

SM: Is not there a very low and finite number of particles in the standard model?

LikeLike

December 13, 2011 at 9:23 am |

This is easy to answer. My observation is not about labelling numbers with different types of particles. It is just about labelling them. Actually, come to think of it, it’s a good question: one can use only fermions, because bosons tend to clump together. Thus this reinforces my argument, as there are fewer bosons, the field paricles, than fermions, the particles of geometric shape.

LikeLike

December 13, 2011 at 9:25 am |

What prevents me to go to G, the largest number, and just add one? Why can I not just think about it? What does that have to do with elementary particles?

LikeLike

December 13, 2011 at 9:29 am |

Any symbol uses at least one elementary particle, a fermion. If one says that ideas do not have to be labelled with a fermion, one says ideas are not material, they are literally out of this world. One believe in the non physical spirit sitting in us. That is in contradiction with the general contemporary mood of science.

PA

LikeLike

December 14, 2011 at 6:44 am |

Oh dear. I leave the conversation for one second and the ‘last fermion’ pokes its head into the stew.

Shouldn’t it be clear by now that science should icks-snay on the finite-article-pay thing. The infinitely dividing towards zero game should at least have given you pause, since its physical properties must therefore match its mathematical properties in order for the game to be valid. Is the fermion the absolute last particle or point of existence we can fundamentally conclude or is it the last particle our current form of technology can conclude? The light-year equivalent of outward physical distance has a comparative inward physical distance as well, and fermions are the equivalent of maybe scratching the edge of our solar system, hardly definitive.

LikeLike

December 14, 2011 at 7:24 am |

Michael: If I understands well what you are saying, you insinuate that information could be stored at a sub-fermionic level (perhaps, maybe). If that were possible, and

if it were be possible to make ever smaller symbols with yet to be imagined substructures, my reasoning would indeed fail. But that seems impossible to me. Even if superstrings, or supermembranes existed, my reasoning would still work. And nobody has even imagined anything smaller.BTW, I never spoke of a “last fermion”, just that one fermion at least would be needed per symbol, at least.

PA

LikeLike

December 17, 2011 at 12:36 am |

Who said anything about storing information? Where does an ‘apple’ fit the word used to describe it? It doesn’t, it’s too busy being an apple to consider a symbol for itself. Existing and having a ‘space’ to store some measure of extraneous information linked to it is a bit… pendantic (for lack of a better word, one will come to me later.)

The word follows the object, not the other way around. The symbol is the shadow, the object itself the thing that obstructs the light. By simply existing, an object must define itself, creates its own symbol that allows it to be translated. What came first, the Higgs particle or Peter Higgs?

LikeLike

December 17, 2011 at 10:31 pm |

Dear Michael:

I would keep a very wide path away from the Higgs: 6 people claim to have invented the concept. Besides, elementary particle physics and QFT do not know what is meant by “elementary particle”. More on this soon…

Now for your point.I agree with some of what you say. However you still do not get my main point. For you a symbol is made of? Of what exactly? A “shadow”? I tangled with exactly that point with some top young research mathematicians a few days ago. Just like you, they did not understand my main point. My point is that symbols do not sit in another universe. There is no other universe: Brian Greene style physics is sheer BS.

So to each symbol at least one (and of course much more) particles will have to assigned, just so that the symbol be. Symbols are made of stuff, of something, and that means, of elementary particles. And there are only so many to go around.

Symbols, and more generally ideas, are physical, that is why there is a largest number.I let my math friends sit on it. I am calm. They will come around. I know the whole world is different now. All of math will have to be finitely redone. Or then have to learn to use the geometrical, connected logic I alluded to.

PA

LikeLike

December 18, 2011 at 11:41 am |

If I called it the “unknown particle,” would you have known what I was talking about? Splitting hairs about who theorized about what and when, in any discussion relating to ideas that have to have some name attached to them, assumes that I am the one naming the Higgs particle. It is what its called in the majority of the scientific community, for the moment, and since I am not going to footnote every scientific idea I produce, a certain amount of shortcuts needs to be assumed. I am fully aware that science changes all the time (bye bye Planet Pluto *sniff*) and until a definitive answer is given to the “Higgs” particle, I will continue to call it such.

Ok, I have a glimmer of what you are attempting to grasp at, but its sitting at the end of some very pretzely logic. I will rephrase my idea and then see if I understand yours and why I see a conflict. For me, the object apple could very easily sit in the middle of nothing without being named. Because it exists by itself, it has no symbol. Symbols, in this sense, exist only as the result of differentiation. A worm enters, stage right, and crawls up to the apple and says “You are an apple and I am a worm.” The symbol “apple” then exists with the worm, in much the same way the symbol “worm” exists with the apple because we basically symbolize what we aren’t. Descartes’ Man has to think before he can know himself abstractly.

But lets say the worm dies. There is no consciousness left to speak of. The apple is once again sitting in the middle of nothing. By your reckoning, the symbols for both apple and worm have died because the thing that allowed for them to be symbols has also died/ended. The apple does not know itself to be a symbol, it does not know any symbol, the symbolizer, the creator of symbols is no more, which still leaves an apple sitting in the middle of nothing. But is there an apple sitting in the middle of nothing. The apple has no consciousness. It does not know where it came from, it does not know where it is going. The only thing that did, the worm, died. But what then is sitting in the middle of nothing? It can’t be an ‘apple’ because that would be, by your theory, outside of itself. You then conclude there is no ‘apple’ sitting in the middle of nothing because the only thing that would contradict you, the worm, is dead. If there is no apple, there can be no worm to interact with said apple… and down the river of De Nile we go.

You can exist without speaking, but you can’t speak without existing. Going silent means you have ceased making sounds. But in order to cease making sounds, you had to have made sounds. But before you made sounds, you were silent. Before there were symbols, there were no symbols, but if there were no symbols, where did symbols come from? You are looking for something you are fundamentally denying without realizing it, and then wondering why you can’t find it. Reminds me of the Woody Woodpecker cartoon when he stares into the bowl of soup and keeps repeating the sentence he reads “I can’t see a thing.”

LikeLike

December 18, 2011 at 2:32 pm |

Michael: You use some interesting ideas and formulations. I will pass on the fact you assume I deny without realizing… That would be really ironical, as

denying denial with realism is my main philosophical thrust.The conversation we are engaged in, is, in some ways, as old as Plato. It got really hot about 1,000 years ago around Paris. There are many concepts, entire theories, and all variants in between, attached to it: nominalism, realism, the problem of universals, conceptualism…

But I claim definitive progress has been made. Why? Because of elementary particle theory, Quantum physics, the theory of gravitation, plus the neuronanobiology around the corner. It was just a matter of putting it all together. Which only someone having seriously studied those fields could confidently do, let me immodestly add. I am not mentioning this for my personal glory, as I don’t exist as a person, but to explain that this obvious idea was not found before, simply because no one had such a deeply diverse background (and I had the good fortune to interact with many of the greatest). Not too many have studied

Model Theoryand theTransplanckian Scale. In depth.I agree with your sentence: “You can exist without speaking, but you can’t speak without existing.” However what is speech, at the most basic level? A physical effect, a dynamic with an indentifiable pattern relating to previous pattern. The most fundamental notion here is “physical effect”.

I am just saying, first and fundamentally, that, just as speech is physics, so are ideas.

A speech is a pattern of elementary particles evolving in time. Could one make a long speech to a worm by just moving, say, ten elementary particles? Sure. The worm would play the role of the tape in the Turing machine, become the tape. Could we make a long speech with ten elementary particle, and no worm to register it? No. But one could make a long speech if one had trillion elementary particles at one’s disposal. Even without any worm. The speech would become the worm. And one needs many particles to make a worm.

Worms are made of zillions of elementary particles. But one trillion zillions is still a finite number. It is precisely because a worm is made of a finite, but very large number of elementary particles that it can store a very long discourse. But there is a limit to storage. A limit to speech.

We are the worms. However we have only a FINITE number of elementary particles at our disposal to geometrically arrange in ourselves. That FINITE number is given by considerations in gravitation and the transplanckian scale. (So, indeed, by studying the same field of knowledge I came to the conclusion that Special Relativity broke down at very high energies, causing Faster Than Light effects.)

To arrive to my conclusion about symbols, one needs just assume the ultimate realism:

ideas are physical objects, just like speeches are physical objects, or caravans in the desert are physical objects.PA

LikeLike

December 18, 2011 at 5:05 pm |

I do not doubt whatever academic and/or professional credentials you have accumulated, but if the thrust of your argument boils down to “I’m a professional and you aren’t,” it undercuts your argument to begin with. Being a professional does not mean you can’t be wrong.

Again, we argued this point previous with the finite tacos metaphor and I agreed with you then. But you are extending the metaphor to the point where it contradicts the basic laws of physics itself. And I say that because whatever number you are attempting to arrive at can only ever be conditional. In other words, you could measure the number of tacos I have eaten in my life and that number would be ‘finite’ but it wouldn’t ever be authoritative because it assumes certain conditions that don’t have to be true. I could have eaten a taco when you weren’t paying attention, and all it would take would be one unaccounted for taco to upset your number. But according to you, that could never happen because A] already has it covered. But it only has it covered because it remains undefined. Once it is defined, all it would take would be one extra ‘push’ to make it irrelevant. Even if you told me, on my deathbed, that I was about to eat my last taco, I’d do my best to shove another one in my mouth, but the very act of doing that, even if I did not succeed, would jeopardize your number for how could I even attempt a number beyond it? Even if I don’t eat another taco, I can still hold one in my cold dead hand. I have not ‘eaten’ the taco, so it doesn’t change that number, but what is that thing in my hand, a poodle? The property that defines the last particle had to be ‘true’ for the previous particle in order for it to lead to that last particle, and the ghost of the particle beyond the last is still sitting there in the dark because of what made the two previous particles true to begin with.

If I can start, I can stop. If I can stop, I can start. Infinite stopping would contradict the possibility of starting to begin with.

LikeLike

June 19, 2014 at 1:34 am |

[…] (To excuse Descartes, the notion of countability had not yet been clearly defined in his time; it leads, in turn, to the finiteness of speech, modulo my finite mood.) […]

LikeLike

July 9, 2015 at 9:22 pm |

[…] pure logic which interface strongly with mathematics.) I have proposed to go much further with a different philosophical insight in Number Theory (in still another direction Archimedes could not have […]

LikeLike

December 18, 2015 at 6:18 am |

[PA sent to Quora Are there mathematical arguments against the existence of infinity? https://www.quora.com/Are-there-mathematical-arguments-against-the-existence-of-infinity/answer/David-Joyce-11 ]

***

Excuse me, how to you construct “infinity” from the usual axioms? Or a “potential infinity”? How can one “assume” something, “infinity”, that one has not defined? Absent a construction of “infinity”, or “potential infinity”, it cannot lead to contradiction.

Nor can introducing “infinity” as an axiom lead to a contradiction.

Another objection is that, indeed, assuming “infinity” enables to do some sort of mathematics. However, absent “infinity” another type of mathematics would have to be practiced.

Maybe “infinity” dependent mathematics is a specialized type, similar to making Euclidean geometry… infinitely: something that can be done, but something nobody does anymore.

To justify “Infinity” with the “one more” argument, assumes that there is an infinite amount of energy to keep on writing… “one more”. This is an axiom not found in nature. Nor in Euclid’s axiom (so his well-known “proof” is false! Just observing everybody says it, does not make it true!)

Axioms of mathematics were always written as an abstraction of nature.

LikeLike

July 1, 2016 at 2:57 pm |

[Sent to Quanta Magazine, July 1, 2016.

https://www.quantamagazine.org/20160630-infinity-puzzle-solution/ ]

The idea of infinity was invented this way: if n is the largest number, (n+1) is larger. However, this assumes that, having used n symbols, we can find still another. So what are the “symbols” going to be? If they are fermions, there is clearly a finite number within the event horizon. Thus there is a number n so that (n+1) does not exist.

https://patriceayme.wordpress.com/2011/10/10/largest-number/

LikeLike

July 1, 2016 at 11:51 pm |

[…] “proven” by the simplest reasoning ever: if n is the largest number, clearly, (n+1) is larger. I have long disagreed, and it’s not a matter of shooting the breeze. Just saying something exists, does not make it so (or then you believe Hitler and Brexiters). If I […]

LikeLike

July 2, 2016 at 3:33 pm |

[…] “proven” by the simplest reasoning ever: if n is the largest number, clearly, (n+1) is larger. I have long disagreed with that hare-brained sort of certainty, and it’s not a matter of shooting …. (My point of view has been spreading in recent years!) Just saying something exists, does not make […]

LikeLike

December 1, 2017 at 8:56 pm |

[…] https://patriceayme.wordpress.com/2011/10/10/largest-number/ […]

LikeLike

January 11, 2018 at 9:32 pm |

[…] after a while, as any computer knows, and I have made into a global objection, by observing that, de facto, there is a largest number (contrarily to what fake, yet time-honored, 25 centuries old proofs pretend to […]

LikeLike

January 5, 2022 at 9:32 pm |

In the argument the author limits himself with a closed system “the observable universe” while also admitting there is more beyond the observable, so no doubt in a closed system there are limits, but who is to say the totallity of all is a closed system?

LikeLike

January 5, 2022 at 9:34 pm |

Amos Kissel: That the universe may be truly infinite (it is de facto so at this point), so we may not run out of quanta to represent numbers (which is my original argument) is a cogent objection that you make. Indeed. However,

physics presently assumes, the universe to be finite, locally, thanks to RENORMALIZATION. Thus existing QFT validates my argument, ironically enough… OK, more later, thanks for te comment! 😉LikeLike

April 14, 2022 at 8:02 pm |

Infinity not solved yet in 2022:

https://www.newscientist.com/article/mg25433822-900-infinity-has-long-baffled-mathematicians-have-we-now-figured-it-out/

LikeLike