Adopting an ultrafinitist system leads you to several conundrums not faced with infinitism. Here are two :

1. The real numbers no longer form a continuum

If $N_{max}$ is the largest possible number, then $\frac{1}{N_{max}}$ must be the smallest (Otherwise, if there exists some k such that $k\lt\frac{1}{N_{max}}$, then $\frac{1}{k}$>$N_{max}$, a contradiction). This smallest number in turn implies that the set of reals is discrete (i.e., no more irrationals). Thus, the diagonal of the square is now commensurable with its side, which we've known to be untrue since the time of the Pythagoreans! (Also, points have size!)

2. Series can be expressed that have no (acceptable) sum

Consider the series 1 + 2 + 3 + 4 + ... If the last term is $N_{max}$ (which should be allowable), then the series sums to a quantity greater than $N_{max}$, a contradiction.

For these reasons (and others) ultrafinitism is inadequate to describe modern mathematics.

Oh, I'm no finitist. As I've pointed out, finitism (or worse, ultrafinitism) leads to some odd results : you have to truncate $\pi$ (which turns circles into polygons), you have to deny irrationals, you destroy the foundations for calculus, lines no longer consist of an uncountably infinite set of points, etc.

You may be onto something about personalities. I think it's also possible that one of the sources for the OP may have been some half-remembered quote such as the one from the NASA engineer, above. But perhaps I'm being too generous.

Another interesting discussion topic (and perhaps what the OP was alluding to) would be the distinction between pure math and applied math. And the usefulness of the former.

And now in support of the much-maligned OP - a quote from Marc Rayman, a chief engineer at NASA (with the coolest name any employee of NASA ever had) :

Let's go to the largest size there is: the visible universe. The radius of the universe is about 46 billion light years. Now let me ask a different question: How many digits of pi would we need to calculate the circumference of a circle with a radius of 46 billion light years to an accuracy equal to the diameter of a hydrogen atom (the simplest atom)? The answer is that you would need 39 or 40 decimal places. If you think about how fantastically vast the universe is — truly far beyond what we can conceive, and certainly far, far, far beyond what you can see with your eyes even on the darkest, most beautiful, star-filled night — and think about how incredibly tiny a single atom is, you can see that we would not need to use many digits of pi to cover the entire range.

Agreed that many of our TPF worthies misunderstand what is meant by $\infty$. But as a minor defense, we often do get sloppy with our use of this symbol. In calculus classes, for example, when a limit evaluates as $\frac{\infty}{\infty}$, we tell students to try applying L'Hopital's rule. I doubt AS has this usage in mind.

What surprises me about our math-phobic friends on TPF, is that philosophy majors usually love the esoteric. You would think they would revel in knowing more about mathematics than the Great Unwashed. Instead, they make up their own notions and denigrate 5,000 years of developments by some of the greatest minds that have ever lived. Weird.

I gotcha!

$\frac{5}{5}$=1

and

$\frac{8}{8}$=1.

What about $\frac{0}{0}$ or $\frac{\infty}{\infty}$ ? Gotta be 1, right?

0 and $\infty$, rather 1ish!

A little advice, friend : get ya some math learnin'

Now the nature of infinity is an interesting topic to explore!

I have much empathy for your position. Since our lives are finite, it seems impossible to experience an infinite set. But it may surprise you to know that many mathematicians today believe that actual infinite sets exist in math!

You're correct, of course. I apologize to AS.

I now realize that AS is asking for something even more restrictive than the ultrafinitists. Correct me if you disagree :

Finitism rejects the existence of an actual infinite set (sometimes, as in the case of Hilbert, allowing for a potential infinity). Ultrafinitism rejects the existence of very large integers (sometimes positing that only what is constructable in this universe, or by humans, exists) - and by extension, irrationals. AS rejects any number that is not useful to humans, such as Pi beyond a few decimal digits. Ultra-ultrafinitism?

My bad.

Why am I nuts?

Ultrafinitism is usually defined as the belief that really large FINITE numbers do not exist because of constructive limits - what is physically realizable in the universe. And what does the OP posit? That there is a maximum number needed to describe the universe. Anything bigger doesn't need to exist. Did I get something wrong?

When I said "too clearly an example of ultrafinitism to be an accident", l was implying that these ideas most likely originated from an ultrafinitist source. I suppose you could arrive at such a conclusion on your own, but not if you've taken math at university, or even high school. The opposite is taught in math today. (Isn't this why you keep admonishing AS to get some learnin' on the subject?)

Not telling you anything you don't know, but adopting ultrafinitism leads to some odd results

• Pi must be truncated, and becomes rational. Thus the circle cannot be closed, and circles should be open to squaring.
• Irrational numbers don't exist, and thus lines do not consist of points.
• Limits cease to exist, and rhe epsilon-delta definition for the derivative is removed.

Yep. Just trying to help you counter the three-headed anti-modern-math beast.

The OP is too clearly an example of ultrafinitism to be an accident. All protestations aside, I think someone is trolling.

Um, you can google ultrafinitism if you don't know what it is. It is clear the OP is an example of ultrafinitism. I suspect the original poster knew that when they started this (and has been laughing all along).

It's an interesting dualism (over here are humans who have reasons, over there is the rest of the universe that does not have reasons) that seems to boil down to "over here is language, over there is no language".

How else do we decide as to who has reasons? I mean this seriously : What should we assume if we meet a space-faring race that we can't communicate with? That they are simply sophisticated tool-using lizards (or mermen, or whatever)? Only humans have a claim on this ill-defined thing?

A number of species seem to recognize themselves in mirrors - bonobos, elephants, magpies. If they're self-aware, do they not have reasons for acting?

The point is, it's one of those poorly defined concepts that we all assume we know. Like saying, "I can't define art, but I know it when I see it." (By the way, it's probably NOT true that elephants can paint - at least not without a lot of cruel training.)

For your edification (after a jokey intro and build up, it gets really fascinating) :

https://waaa.wnyc.org/radiolab_podcast/radiolab_podcast072922_whales.mp3/radiolab_podcast072922_whales.mp3_ywr3ahjkcgo_717274d1b0e3146c987a0aa1743ab9c7_36521884.mp3?awCollectionId=15957&awEpisodeId=1233831&sc=siteplayer&aw_0_1st.playerid=siteplayer&hash_redirect=1&x-total-bytes=36521884&x-ais-classified=streaming&listeningSessionID=0CD_382_250__15df7bbb0187782f2ae19ef2577f07944f2a64bf

Again, the problem I have with Foolos4 is switching between meanings of "is" in a single sentence. You shouldn't say, "3+1 is 4" AND "3+1 is not 2+2" in the same sentence. Either they're both "is" or they're both "is not".

Taking it a bit further : I concede that 3+1 and 2+2 can have different meanings. We teach children that 3+1 and 2+2 are different ways to arrive at 4 (called partitions when we get to advanced math). But once you're above the age of 8 (or so), to hear someone say "3+1 is not 2+2" is going to be problematic. The speaker is then going to have to explain, "Oh, I meant splitting 4 things into 3 and 1 is different from 2 and 2". The speaker can't just say, "Come on, it's obvious!" (or assume the listener will have Frege's notions of sense and reference instantly leap to mind). And the reason is we learn to associate "plus" with addition, and "is" with equal-to when numbers are being used. A better sentence would be, "3 and 1 is not the same partitioning as 2 and 2".

The problem is your definition of "plus", and you won't answer me. To be generous, I think you mean something like "3 things over here and 1 thing over there" when you say "3+1". But that's called partitioning in math.

What we've arrived at is this: sometimes "is" means "is equal to", and other times "is" means "is the same partition as". When you say "3+1 is 4", you mean "3+1 is equal to 4". When you say "3+1 is not 2+2", you mean "3+1 is not the same partition of 4 as 2+2".

My contention - stated in an earlier comment - is you can't switch between meanings in the same sentence. You can't say, "3+1 is 4, but 3+1 is not 2+2", without sowing confusion. No one will understand you. I don't know why you can't see this. It's like saying, "Bill cans peas, but Sally cannot peas". It's nonsense.

So "is" means equal to. Unless it doesn't. I'm sorry, but that's incoherent. If "is" means equal to, then 3+1 is 2+2. If "is" doesn't mean equal to then you need to define it as more than "looks like".

And I ask again, what is your definition of "plus"? Is it commutative? Does it have an identity element? Does it allow for inverses (i.e., negatives)? Is it mathematical at all?

You've evaded many of my questions (or shrugged them off with a "that's obvious!" argument). I want to go back to an earlier question. If you are holding a donut in each hand, is it 1+1 or 2? Why does handing me a donut matter?

Gotcha. But we've moved on from the ancient Greeks' weird take on numbers. Now we accept that one is a number, and so is zero. So are negatives, and irrationals, and imaginaries, and transcendentals. (By the way, "number of things" is a metaphor.)

Math left the Greeks behind a long time ago. Their contributions were incredibly important but eulogizing their achievements sometimes led to misunderstandings or wrong turnings. Look at how clinging to Euclid's fifth postulate held back geometry.

Not every pronouncement by Plato and Aristotle should be held up as exalted. Aristotle thought women had fewer teeth than men.

Finally, I believe very few mathematicians today belong to the intuitionist school of thought.

A math joke to lighten the mood ...

A biologist, a physicist, and a mathematician are sitting on a bench across from an apartment building. They observe two people enter the building. Five minutes later, three walk out. How does each react?

• The biologist : "They must have reproduced."
• The physicist : "My initial measurement must have been wrong."
• The mathematician : "Now if one person enters the building, it will be empty again."

I have been giving this some thought. Our debate has nothing to do with the word "is", it's with the word "plus".

I realized I have no idea what you mean by the + symbol. It could indicate the addition of numbers as in arithmetic (this seems unlikely given your rejection of "is" meaning "equal to"). It could indicate the cardinalities involved in the union of disjoint subsets (although you seem to recoil at the notion of partitions). What is your definiton of "plus"?

Then questions follow

• Is your "plus" commutative? (i.e., are 3+1 and 1+3 the same or different?)
• Does your "plus" have an identity element? (i.e., if you have all the donuts and I have none, is that 4+0 or just 4?)
• Are negatives defined for your "plus"? (i.e., does -1 + 5 have meaning for you? How do you count -1 donut?)

Let me know. Hope you're not too angry.

From the blurb you provided

Specifically at issue is Husserl's expressed concern over the loss of an "original intuition" to ground symbolic mathematical science ...

and

For the Husserl of Crisis, the history of this breakdown consists of ... symbolic algebra, which latter surreptitiously substitutes symbolic mathematical abstractions for the directly intuited realities of the real world ("life-world") ...

What are "directly intuited realities"? Could it include pebble counting? Apparently, abstraction is the devil's work (and thus Kronecker hounds poor Cantor into depression).

If you're not too furious, please check out the next post.

Husserl and Klein want to take math back to pebble counting. And you have apparently joined in. Good for you. I'm not an intuitionist and have no interest.

You object to my 1+1 vs. 2 example. I assume you think you're holding 2 donuts. But why does handing me one turn it into 1+1? What if we are holding the donuts so that they are touching?

If 4 people each hold a donut, you would say that that is different from 1 person holding all four (1+1+1+1 is not 4). But what if 2 of to them live in Paris an 2 in New York? Isn't that 2+2?

If I'm holding a donut and a dollar, the cardinality of the set of objects I'm holding is 2. You seem incapable of accepting a set made up of disparate objects.

So you want to take math back to pebble counting. Okay, let's try a thought experiment. If you hold a donut and someone hands you another donut, do you have 1+1 or 2 donuts? Does holding them in one hand or in separate hands matter?

You're using "is" to refer to the partitioning of sets. And now that I know, I'm fine with it. But we could have avoided any confusion if you had said from the beginning, "2+2 and 3+1 are different because they break up the number 4 in two distinct ways".

You have stated, over and over, that "2+2 is 4" and "3+1 is 4". Without qualifying the "is". Go back and check.

Now it's some great revelation that 2+2 is NOT 4 ?

In math, we call what you're referring to partitions. But unless you and your audience already know that you are talking about partitions, no one - NO ONE - would say "2+2 is not 3+1". Especially after having claimed "3+1 is 4".

Except the mystics on TPF. You're always searching for the woo.

So from now on, when discussing numbers, we know that "is" refers to partitions. Got it.

Oh, and your last paragraph? A total non sequitur.

If 2+2 is not 3+1 simply because they represent different partitions of 4, then 2+2 is not 4 either.

If 2+2 is 4 because they have the same numeric value, then 2+2 is 3+1.

The only way I can make any sense of what you're saying is to assume that you are thinking of "4" as the name of a set to which distinct elements 2+2 and 3+1 belong. But does 4 belong to "4"? Then 2+2 is distinct from 4 and clearly 2+2 is not 4.

As a mathematician, I have to know : why do you think that saying “two plus two is four” will lead to confusion? How might one misconstrue “is”?

There appear to be two uses of “is” in mathematics

• to indicate equivalence (e.g., two plus two is four).
• to indicate that an element belongs to a set (e.g., two is odd).

What else is there?

I breathlessly await your reply.

Wow. I encounter so many people on TPF who do not know basic math, it's striking.

By your logic, if you kept all 4 donuts, that would be different from sharing them out 3 for you and 1 for me. So I guess 3+1 is NOT 4 after all!!!

You are aware that 2+2 = 3+1 ?

Agreed that "2+2" is not the same thing as "4" - one requires three keystrokes, and the other just one. So if "is" means equals (as you say), how can you claim "3+1 is not 2+2"?

You want to find mysticism here. I stand by my claim : you are playing fast and loose with your definitions.

Wait. I think I've got it now. You're thinking of "4" as the name of a set whose elements include "3+1", "2+2", etc. So a better sentence would be, "3+1 is a type of 4".

But then is there another set called "2+2"? What belongs to it?

So you've changed the meaning of "is" within a single sentence. Clearly 3+1 does not look like 2+2, but neither does it look like 4. To say "3+1 is 4" but "3+1 is not 2+2" is incoherent.

No. It's still a category error. "Two plus two is four" clearly implies numbers. "Two" and "two centimeters" are not the same. Adding units (cm, mm) changes the sentence.

What is your definition of "is"? (asked Bill Clinton).

3+1 "is" 4 but 3+1 "is not" 2+2 — Fooloso4

Explain yourself. Do you have some special mathematical definition of "is"?

Two centimeters plus two millimeters do not equal four. (Four what?) They are just equal to "two centimeters and two millimeters". — Alkis Piskas

Why introduce this non sequitur? "2 + 2" has nothing to do with "two centimeters and two millimeters". It's like saying "red and yellow make orange" must be false because "red firetruck and yellow car don't make orange anything". Category error.

If you still want to introduce mysticism into math, then what do you do with the sentence, "two centimeters and two millimeters is four units of length measurement"? Seems OK to me. So "2 + 2 is (still) 4".

(Next we'll have TPF worthies jumping in to claim zero is not a number, and lines are not made of points.)

Some suggestions :
A) non sequitur
B) deflection
C) being a member of the Republican party

:up: Thanks

Had an exchange with Pie about this. As you are obviously aware, Euler was one of the greatest minds mathematics has ever known, but infinite sequences and series were not always his friends. I'm not an Euler scholar, though I have several works discussing his many contributions. I must have missed this one.

:up: re the math symbols.

What surprises me is that Euler was so far ahead of his contemporaries in most areas, but seemed to have weird blind-spots from time to time.

I still don't see the confusion over negatives and their operations, but then I "do" math every day. Oftentimes familiarity makes it difficult to see how others must view the same. Thinking about this particular topic, one of the clearest explications (for me) comes from business - accounting to be precise :

It's well-known that profit = revenue - cost (clearly, revenues are positive, costs are negative). So finding new revenue is positive for a company (adding a positive is positive). Losing revenue is bad for profit (subtracting a positive is negative). Adding a cost is also bad (adding a negative is negative). Finally, removing a cost increases profit and is positive for the company (subtracting a negative is positive). Removing multiple costs (multiplying negative by negative) is also a positive.

I don't know if it helps, but that's (probably) my last comment on this. Maybe I'll start a topic proving square circles exist. Hmm ...

Found a reference citing the sequence

[10 / (1/n)] as n -> infinity,

with the claim that it eventually becomes negative.

(don't know how to code math symbols, and don't have the time to look it up).

If that sequence is due to Euler, it's a classic case of Euler's lack of rigor regarding infinite sequences and series. Yes, as n -> infinity, 1/n -> 0 and the sequence approaches infinity.
But n -> infinity does NOT imply that 1/n will become negative at some point. That's not the way limits work! 1/n never becomes negative, and so the sequence itself never becomes negative.

What stories like this point out is that there was a considerable lack of understanding regarding infinite sequences and series at that time. Even so great a mind as Euler's made a hash of it. And the nonsense continues to this day. A claim that infinity = -1/12 was made recently on this very forum. In tracking down the source, I came across an explanation that started with the ridiculous statement that
1 - 1 + 1 - 1 + 1 - ... = 1/2. That's insane! I often teach that particular series to Calc II students as an example of a divergent series. We should know better now

Euler, in the latter half of the 18th century still believed negative numbers were greater than infinity. — jgill

Can I ask you where you got this from? I know Euler played fast and loose with infinite series, but I can only find this bit about negative numbers mentioned on an obscure Wikipedia comments page. Since Euler is one of the greatest minds mathematics has ever seen, this seems like an odd mistake.

Sorry, I'm done. A basic math course might help.

Then again, it might not.

I suppose it's a good thing that folks see math as "poetry" and such. But negative numbers? Really? Did you just discover them last week?

Cognitive scientists believe that children are ready to learn negative numbers by the age of 11 or 12. That's sixth grade in US schools.

There are a host of websites dedicated to explaining multiplication of negatives to children of this age. The very first one that came up when I googled was something called "How to Adult". Here are a few ways they explain it :

When I say "Eat!" I am encouraging you to eat (positive)

But when I say "Do not eat!" I am saying the opposite (negative).

Now if I say "Do NOT not eat!", I am saying I don't want you to starve, so I am back to saying "Eat!" (positive).

I like this one :

The tank has 30,000 liters, and 1,000 liters are taken out every day. What was the amount of water in the tank 3 days ago?

We know the amount of water in the tank changes by −1,000 every day, and we need to subtract that 3 times (to go back 3 days), so the change is:

−3 × −1,000 = +3,000

The full calculation is:

30,000 + (−3 × −1,000) = 30,000 + 3,000 = 33,000

So 3 days ago there were 33,000 liters of water in the tank.

I guess if it tickles you to contemplate negatives all day, have at it. There are just so many deeper notions in math to ponder, it seems a bit silly to me.

You may want to try a better example.

ALL living things consist of cells. But how many varies from beastie to beastie. Some consist of one cell, some consist of millions. The term "cell" by itself has nothing to do with being a living organism.