An object that has no members is either the empty set or an urelement. And of course, an object that has in it only one object is a non-empty set. — TonesInDeepFreeze

An "object" is one. To allow that an object has members requires a special definition. We can say that this object is a set, and define a "set" as a collection of objects. But to say that there is a collection of objects with no objects is contradictory, whether you admit to this fact or not.

It seems that you do not recognize the fundamental distinction between one and many, and you are now trying to reduce many to one, by saying that a set is an object. One is not a plurality, and a plurality is not one. That is a basic self-evident truth. If you want to talk about a category, or class of objects, this is something completely different. We cannot name the category "an object", and also name the members "objects", without equivocation. Do you apprehend the category mistake, and consequent fallacy of equivocation which occurs if we call both the category, and the members within that category, by the name "object"?

Talking to you is like conversing with a computer. — Agent Smith

I thought for a while, that Tones was a bot, endlessly repeating the same thing over and over without grasping the logical problems with what was repeated. Then I realized that this is simply the way mathematics is. The students are taught very specific principles, and their minds are funneled down a very narrow path, You could say that they are programmed, like a computer is programmed, and the possibility that the program is not a very good program, so that they are being misled, is excluded from the program. The students are discouraged from looking outside the program, and seeking the truth, like philosophers do, because truth is not important to mathematics.

On point!

It hasta make sense and (rigorous) logic is all about that - hats off to @TonesInDeepFreeze - making sense (of the cosmos).

From a psychological perspective, I wonder if the desire to reach infinity is based on a feeling of something not being good enough in ourselves, or our experience.

I seek more and more, with the idea that more will eventually result in completely full. More knowledge, more experience, more understanding.

But more doesn't lead to 100% full, or so I've been told

The more that is had, the bigger becomes the comparison of measurement
So that even if had the whole earth, I would end up comparing my little earth to the universe and feel quite poor.

Instead, aescetics have told me through writings, that I have to become 100% empty. Only a 100% empty mind has room for infinity.
As motivational coach Tyler Durden of 'Fight Club' put it, "It's only when we lose everything that we are free to do anything."

And:
"When the doors of perception are cleansed, we see things as they really are, infinite" - One of those classic poets, William Blake, I think. Also what the band, The Doors, based their band name on, or so I've heard.

Reality ultimately can't be limited, because limitation requires two. One thing limiting another.

I also reminded of an interpretation of the symbolism of the tower of Babel.
They tried to build a tower tall enough to reach God.

Is that we are are doing with the intellect? Trying to attain Infinity through building conceptual realities?

Fun to think about. Infinity literally is the one thing that can't ever become boring, since things only become boring once we understand everything about them!

Now consider the fact that in a universe that's finite there's gotta be a number that is the upper limit of a counting processes that yields the largest number possible/required to describe this universe — Agent Smith

The universe is physical. Numbers are not. They are abstract objects. Whether the universe is finite or infinite has nothing to do with numbers being finite or infinite. Their nature is totally different and their existence is of a different kind, too. The only relation between numbers and universe that I can see is the following: numbers are representations of things in the universe, considered individually or in groups, categories etc. When I say "5 (five) apples" I refer to "5 (five) individual objects" or "a group of 5 (five) objects". Both the numeral "5" and the word "five" are symbols that represent a "quantity", which is also another abstract word, a concept.

So, if the universe is finite, it means that it contains a finite number of elements, such as atoms (microcosm), stars (macrocosm), etc. If it is infinite, then it means that it also contains an infinite number of elements.

Now, we also have irrational numbers, like PI, which have a sequence of decimals that looks infinite. But this is theoretical and has nothing to do with whether the universe is finite or infinite.

Finally --but not the last-- we talk about the magnitude of physical objects, which is also an abstract idea. For example Plank's constants refer to the smallest and largest number calculable magnitudes, which range from 5.4 x 10^-44 ("Planck time in seconds, the shortest meaningful interval of time in seconds, and the earliest time the known universe can be measured from" and 5.1 x 10^96 (Planck density, the density in kg/m^3 of the universe at one unit of Planck time after the Big Bang). (Re: https://www.physicsoftheuniverse.com/numbers.html)
But I don't think that these can be used as a proof that the universe is finite. Well, my knowledge of Physics is quite limited to be able to judge.

@Yohan

Nice angle, the psychologist's view on $\infty$. Speaking for myself, I find the expression "I love you infinity" very thought-provoking and awe-inspirng. Also in the same category is the answer "11" to the question "how's the pain on a scale of 1 to 10?"

@Alkis Piskas

:ok: It's just that I wanted to know if scientific calculations invovling infinities could be tamed in a manner of speaking.

It's just that I wanted to know if scientific calculations invovling infinities could be tamed in a manner of speaking. — Agent Smith

Renormalizations :roll:

Renormalizations — jgill

Whazzat?

:scream:

Renormalizations & Regularizations

Saved for later! Danke!

the observable universe is far too small to contain an ordinary digital representation of Graham's number, assuming that each digit occupies one Planck volume, possibly the smallest measurable space. But even the number of digits in this digital representation of Graham's number would itself be a number so large that its digital representation cannot be represented in the observable universe. Nor even can the number of digits of that number—and so forth, for a number of times far exceeding the total number of Planck volumes in the observable universe.
[snip]
At the time of its introduction, it was the largest specific positive integer ever to have been used in a published mathematical proof. The number was described in the 1980 Guinness Book of World Records, adding to its popular interest. Other specific integers (such as TREE(3)) known to be far larger than Graham's number have since appeared in many serious mathematical proofs, for example in connection with Harvey Friedman's various finite forms of Kruskal's theorem.

https://en.wikipedia.org/wiki/Graham%27s_number

Don't go messing' with mathematicians with your "c'mon, be reasonable" attitude; they'll have none of it.

Merci beaucoup for the warning!

Graham's number appears in a mathematical proof but what about calculations that use actual measurements such as mass, velocity, charge and so on? Infinity appears in black hole calculations and once that happens, the results evoke the response nec caput nec pedes. I only wanted to explore the possibility of some kinda workaround to this rather lamentable state of affairs in the math of physics.

Initially I was of the opinion that infinity had to be replaced by an extreme number like Graham's number but the result $\sum_{n=1}^{\infty} = -\frac{1}{12}$ (used in string theory) is evidence the number that we swap infinity with in a calculation, surprise, surprise, doesn't have to be large, a wannabe infinity; a small, nevertheless most special number like $-\frac{1}{12}$ will do just fine.

Don't go messing' with mathematicians with your "c'mon, be reasonable" attitude; they'll have none of it. — unenlightened

I guess that's why TIDF split the scene. But we were just playing, it's nothing harmful.

Initially I was of the opinion that infinity had to be replaced by an extreme number like Graham's number but the result ∑∞n=1=−112∑n=1∞=−112 (used in string theory) is evidence the number that we swap infinity with in a calculation, surprise, surprise, doesn't have to be large, a wannabe infinity; a small, nevertheless most special number like −112−112 will do just fine. — Agent Smith

Fractions are actually very tricky. The common approach is to assume that any object can be divided in any way, so there is an infinity of possible divisions for each thing to be divided. In reality though, the way an object can be divided is highly dependent on the composition of the object. So producing ratios in theory, and applying them without proper standards as to the true way that things can be divided, can be very misleading.

This becomes quite evident in wave theory, as the ancient Pythagoreans who studied harmonies and the properties of musical tones found out. There is a very real problem with the assumption that time (hence frequency) can be divided arbitrarily (in any way that one wants). This produces the uncertainty principle of the Fourier transform.

Well, that went over my head, but what caught my eye is that what you seem to be saying is our options aren't $\infty$ i.e. we may restricted to finite mathematics, even if not all the time, some of the time.

the observable universe is far too small to contain an ordinary digital representation of Graham's number, assuming that each digit occupies one Planck volume, possibly the smallest measurable space. But even the number of digits in this digital representation of Graham's number would itself be a number so large that its digital representation cannot be represented in the observable universe. Nor even can the number of digits of that number—and so forth, for a number of times far exceeding the total number of Planck volumes in the observable universe.

That last part... holy shit!
:strong: :wink:

Calculations in physics. The Lorentz factor is unbounded. — jgill

The equation is unbounded. But is it unbounded in this universe? We cannot accelerate an object arbitrarily near to c, since that requires arbitrarily high amounts of energy, which is not available.

Our friend, AS, inquired about an upper bound on numbers used in physics formulas. I assume both theoretical and practical. If strictly practical perhaps the inverse of one of Planck's constants.

Late to the party as always.

True, numbers are infinite. — Agent Smith

False actually. I cannot think of a single number that is, let alone all of them.

That aside ...

In all likelihood there's a number that would make us go "Yeah, this is it! It all makes sense now!" and that number is probably going to be unimaginably large but finite. — Agent Smith

If that number causes it to all make sense, it probably isn't unimaginably large.

I read somewhere that the observable universe contains roughly 10^80 atoms. That should be a good place to start at least when it comes to matter, oui? — Agent Smith

Perhaps a reasonable place to start when talking about large useful numbers.
Now consider all the possible ways to arrange all those atoms in a given visible universe volume. This gives all the possible states of a visible universe, which is quite useful for positing such navel-gazing concepts as "How far is it to the nearest exact copy of me?". That's a really huge but still useful number, but it starts with the number you gave above.

Tegmark did such a calculation and got a number something like 10^10^28 meters away (quick google search) which is smaller than your random useless number above, but still much bigger than the 10^80 one.

Tegmark doesn't read his own book. A better answer is somewhat closer by if the whole book is taken into consideration.

PS: I couldn't figure out how to do the superscripts like you have in your posts. Maybe it's buried in the terse menu above, but I can't find it.

Well, that went over my head, but what caught my eye is that what you seem to be saying is our options aren't ∞∞ i.e. we may restricted to finite mathematics, even if not all the time, some of the time. — Agent Smith

What I think, is that we allow "infinite" so that we will always be able to measure anything. If our numbers were limited to the biggest thing we've come across as of yet, or largest quantity we've come across, then if we came a cross a bigger one we would not be able to measure it. So we always allow that our numbers can go higher, to ensure that we will always be able to measure anything that we ever come across. In that way, "infinite" is a very practical principle.

:ok:

& @jgill

I wonder what Max Tegmark would've said about my query.

I'm neither a mathematician nor a physicist so you'll have to cut me some slack here. Imsgine you're doing a calculation on black holes and you end with $\infty$ in your result.

Last I checked there are many infinities, each larger than the other. the "smallest" of them being the infinity of natural numbers viz. {1, 2, 3,...} aka $\aleph_0$. Are all the infinities that appear in physics calculations $\aleph_0$?

Are all the infinities that appear in physics calculations [aleph naught]? — Agent Smith

Infinities can appear at singularities. This is the notion of infinity as unboundedness and/or non-infinite but bizarre behavior at such points. What you are talking about are ways of categorizing infinite sets in set theory. The terms of the series S=1+2+3+... form a set having cardinality aleph naught, but the series is unbounded. If Tones tunes in he could explain the details. In my research I never needed anything more.

Why don't you send Tegmark a question about you OP? Then report back. :cool:

Infinities can appear at singularities — jgill

Yea, that much I know; fell in love with that word since I was, what?, 17!

Anyway is the $\infty$ in a singularity physics calculation $\aleph_0$?

perhaps the inverse of one of Planck's constants. — jgill

Now we're talking!

Imsgine you're doing a calculation on black holes and you end with ∞∞ in your result. — Agent Smith

Yea, but that's not a number, so it doesn't answer your question about the largest number we will ever need.

Are all the infinities that appear in physics calculations ℵ0ℵ0?

They're not 'counts' of things, so the question doesn't apply. Those infinities just mean that the equation fails to describe the physical situtation. So for instance, it take infinite coordinate time for a rock to fall through the event horizon. That just means that this choice of coordinate time is singular there, so it cannot meaningfully describe the rock falling through. It doesn't mean the rock doesn't fall through, or that anything even particularly different happens to it there.

It doesn't mean the rock doesn't fall through, or that anything even particularly different happens to it there. — noAxioms

It means that the concept of a rock falling through the event horizon, is incoherent. In other words, the theories applied, mathematics applied, or both, are faulty, because they produce an incoherent scenario.

I dunno how to say this but $\infty$ erects an opaque wall between the equations and the stuff they (are supposed) to describe. My suggestion to physicists/mathematicians who deal with this kinda problems as a matter of routine is to find a workaround, one which to my reckoning involves substituting $\infty$ whenever and wherever it occurs with an appropriate finite number; this could really be a big help in my humble opinion.

It's as if some phenomena that are said to be real are off-limits, no-go red zones; our most powerful tool - math - is rendered useless. Intriguing possibilities emerge, one of which is is Max Tegmark who believes the universe is mathematical right?

There are 10^183 planck length unit volumes in the universe. Might this be nmax?

10^183 — hypericin

Good one! You seem to be on the right track - take the smallest physically meaningful quantity and check how many of 'em fit in the (observable) universe! You get points for being systematic.