X

[Monroe]

No, I suppose they don't. We try to give something a name, and then apply it to other similar things, when no two things are truly the same. Thus, numbers can't exits naturally, since there isn't more than one of anything in reality. I suppose 'one' can exist, but it loses its point when it's the only number.

[Dantius]

Originally Posted By: Monroe

I suppose 'one' can exist, but it loses its point when it's the only number.

One, is the loneliest number that you'll ever do...

That had to have been a setup. I'm sure of it. Then again, I may be the only one who got that reference. *sigh*.

[Shaper Cirikci]

You are not alone in your reference getting.

[Randomizer]

I thought of it too before scrolling down to your post.

 

According to the Bible, Adam's job was naming things.

[Slawbug]

Originally Posted By: Celtic Minstrel

the concept of number comes out of quantity and counting... So, numbers may not manifest themselves physically in reality, but they arise naturally from reality simply by counting.

Originally Posted By: CRISIS on INFINITE SLARTIES, earlier in the thread

what the heck does "arise naturally" mean?

[VCH]

Originally Posted By: CRISIS on INFINITE SLARTIES

Originally Posted By: Celtic Minstrel

the concept of number comes out of quantity and counting... So, numbers may not manifest themselves physically in reality, but they arise naturally from reality simply by counting.

Originally Posted By: CRISIS on INFINITE SLARTIES, earlier in the thread

what the heck does "arise naturally" mean?

Without previous instruction or experience?

[Alorael at Large]

The argument seems to be that we invented numbers by experiencing things that come in numbers. I'm sure that's true, but I don't think that's natural. Ants also experience numbers but can't articulate "one," much less "negative one."

 

—Alorael, who would not blame that entirely on the difficulties with articulating through mandibles.

[Celtic Minstrel]

Originally Posted By: Monroe

Those six apples were, in reality, each a unique entity, or sets of many entities, or a single set as a group. The six (or any other number that one might apply) only exists in our minds as we need apply it.

Yes, each of the apples is unique. Counting isn't necessarily about enumerating items that are identical; it's more about enumerating items that has similar qualities. Each of those six apples is round, with an indent on the bottom and another on the top. Each one has small brown seeds if you cut it open. They may not be the same, but they are similar. (Not in the mathematical sense, but in the sense that they have some properties in common.)

Originally Posted By: Absolutely perfect, peerless PDN

Do nouns arise naturally?

I would say yes; the concept of "noun" is akin to the concept of "object". A neutral observer would see that he is surrounded by objects. Similarly, a verb is an action, sort of an embodiment of change. Even without a predefined conception of verbs and nouns, I think one would notice the distinction between actions and objects.

Other word classes could have similar arguments; adjectives are qualities or properties that an object can possess, for example. Adverbs are a bit harder; I won't attempt an analysis here.

Articles and prepositions, on the other hand, may not arise naturally. After all, there are languages that do just fine without articles. I imagine there could be a language that manages without prepositions, though I do not know of one.

Now, the observant may note that my description of nouns and verbs above does not cover the entire domain of nouns and verbs respectively. Saying that a noun is an "object" is to exclude the possibility of abstract nouns. Saying that a verb is an "action" is to exclude the copula and other similar verbs such as "to seem". But that's okay; perhaps there's no a priori reason for such things to be lumped in as nouns and verbs respectively, but all that means is that we've taken the domains of nouns and verbs and added in some other things, perhaps things that we couldn't quite classify.

Originally Posted By: Monroe

No, I suppose they don't. We try to give something a name, and then apply it to other similar things, when no two things are truly the same. Thus, numbers can't exits naturally, since there isn't more than one of anything in reality. I suppose 'one' can exist, but it loses its point when it's the only number.

It is technically possible to have two completely identical objects, though incredibly unlikely. However, this does not preclude the concept of numbers "existing" in a certain abstract sense, because as I said previously numbers do not measure sameness; they measure similarity.

If I were to take your argument here as valid, does that not mean that there is no such thing as an "apple" simply because no two "apples" are ever identical? Nonsense! We don't call one object an "apple" because it is absolutely identical to another object which we already know to be called an "apple". We call the second object an "apple" because it is similar to the first object; it shares certain properties that we have come to understand as representing what an "apple" is.

Originally Posted By: CRISIS on INFINITE SLARTIES

what the heck does "arise naturally" mean?

It means that there is a natural way to arrive at the concept of numbers with no prior knowledge apart from simply observing the world around you. At the very least, it seems to me that this is the case. Perhaps you can convince me otherwise. Who knows?

Originally Posted By: Absolutely perfect, peerless PDN

Ants also experience numbers but can't articulate "one," much less "negative one."

I'd actually say that the leap from natural numbers (P.S. Could it be that they are called that for a reason? ) to integers is not as natural as the leap from reality to natural numbers. Getting an abstract concept out of counting is one thing; taking away three apples and saying "That's negative three!" is quite a different leap of logic. Obviously there's a way to arrive at that leap, but I don't think it's as "obvious" as "inventing" the natural numbers in the first place.

[The Ratt]

2

 

 

Hey look!!! It's a number!!!

[Lilith]

oh boy let's all argue about mathematical platonism that's sure to be productive

 

here's an interesting data point of uncertain relevance: behavioural experiments have shown that rats can grasp the abstract concept of the number "two"

 

Quote:

However, Russell Church and Warren Meck have shown that rats represent number as an abstract parameter that is not tied to a specific sensory modality, be it auditory or visual. They again placed rats in a cage with two levers, but this time stimulated them with visual as well as with auditory sequences. Initially, the rats were conditioned to press the left lever when they heard two tones, and the right lever when they heard four tones. Separately, they were also taught to associate two light flashes with the left lever, and four light flashes with the right lever. The issue was, how were these two learning experiences coded in the rat brain? Were they stored as two unrelated pieces of knowledge? Or had the rats learned an abstract rule such as "2 is left, and 4 is right"? To find out, the two researchers presented mixtures of sounds and light flashes on some trials. They were amazed to observe that when they presented a single tone synchronized with a flash, a total of two events, the rats immediately pressed the left lever. Conversely, when they presented a sequence of two tones synchronized with two light flashes, for a total of four events, the rats systematically pressed the right lever. The animals generalized their knowledge to an entirely novel situation. Their concepts of the numbers "2" and "4" were not linked to a low level of visual or auditory perception.

 

Consider how peculiar the rats' behavior was on trials with two tones synchronized with two light flashes. Remember that in the course of their training, the rats were always rewarded for pressing the left lever after hearing two tones, and likewise after seeing two flashes of light. Thus, both the auditory "two tones" stimulus and the visual "two flashes" stimulus were associated with pressing the left lever. Nevertheless, when these two stimuli were presented together, the rats pressed the lever that had been associated with the number 4! To better grasp the significance of this finding, compare it with a putative experiment in which rats are trained to press the left lever whenever they see a square (as opposed to a circle), and to respond left whenever they see the color red (as opposed to green). If the rats were presented with a red square - the combination of both stimuli - I bet that they would press even more decidedly on the left lever. Why are the numbers of tones and flashes grasped differently from shapes and colors? The experiment demonstrates that rats "know," to some extent, that numbers do not add up in the same way as shapes and colors. A square plus the color red makes a red square, but two tones plus two flashes do not evoke an even greater sensation of twoness. Rather, 2 plus 2 makes 4, and the rat brain seems to appreciate this fundamental law of arithmetic.

[Jadan]

Not to backtrack, annoy, or even step on anyone's toes, but numbers truthfully have always existed, only being discovered when someone/thing managed to realize where they were.

Take for instance the very first rock ever. Also suppose this rock had roughed edges, giving it something like 18 faces, or Urgh faces. Already that is numbers, or whatever else you wanna call it. Then divide 18/Urgh/Gibber faces by it's two halves = 9, Multiply that by (syllables of Eighteen = 2)+ (Left and Right Hemispheres of Rock = 2) and you arrive at 36.

 

Everything is rife with numbers, and just waiting to be found out. Without a doubt numbers exist, or better it is to say existence is in itself a number.

[Slawbug]

Originally Posted By: Celtic Minstrel

Originally Posted By: CRISIS on INFINITE SLARTIES

what the heck does "arise naturally" mean?

It means that there is a natural way to arrive at the concept of numbers with no prior knowledge apart from simply observing the world around you. At the very least, it seems to me that this is the case. Perhaps you can convince me otherwise. Who knows?

Okay, so it sounds like we are in agreement that, however "natural" you may find the concept of numbers, that concept did not exist until humans came up with it. Quantities obviously existed, but a way to specify them -- numbers -- did not exist until somebody thought of it. So, numbers are "naturally occuring" in the sense that language is natural, not in the sense that trees, rocks, and hedgehogs are natural.

[Lilith]

you know that there's plenty of mathematicians who do literally believe that some or all mathematical entities are things that exist in themselves, independent of the physical world or human thought, right

[Lilith]

i mean i think they're nuts but i'm not a mathematician so maybe i'm not in a position to judge~

[waterplant]

Nouns are products of language. Numbers are products of consciousness - a horse doesn't eat 6 apples. As far as the horse is concerned he or she eats the food that's accessible and most attractive.

 

Originally Posted By: The Ratt

2

 

 

Hey look!!! It's a number!!!

 

Prove it.

[Lilith]

Originally Posted By: waterplant

Nouns are products of language. Numbers are products of consciousness - a horse doesn't eat 6 apples. As far as the horse is concerned he or she eats the food that's accessible and most attractive.

did you read that thing i posted at the end of the last page and if so are you implying that rats are conscious

[Student of Trinity]

Originally Posted By: CRISIS on INFINITE SLARTIES

[N]umbers are "naturally occuring" in the sense that language is natural, not in the sense that trees, rocks, and hedgehogs are natural.

What is the difference between these two senses of 'naturally occurring'?

I'm inclined to say that consciousness is as much a natural phenomenon as teeth on crocodiles or thunder when it rains. So conscious concepts of number are also natural phenomena. Conversely, I think consciousness probably evolved because certain rules really are part of the natural world, and it's an evolutionary asset to be able to recognize and exploit them.

[waterplant]

Originally Posted By: Lilith

Originally Posted By: waterplant

Nouns are products of language. Numbers are products of consciousness - a horse doesn't eat 6 apples. As far as the horse is concerned he or she eats the food that's accessible and most attractive.

did you read that thing i posted at the end of the last page and if so are you implying that rats are conscious

Just read it now - interesting. Rats are clever. But how clever are humans at knowing what an animal knows? At best we can interpret these experiments from a human perspective (a western scientific human perspective at that!).
I will go so far as claim that rats are conscious, but the extent of that consciousness is unknown to this humble waterplant.

[Celtic Minstrel]

Originally Posted By: CRISIS on INFINITE SLARTIES

So, numbers are "naturally occuring" in the sense that language is natural, not in the sense that trees, rocks, and hedgehogs are natural.

I seriously don't see how these are two different senses of "naturally occurring". Just because something was thought up by humans does not make it "unnatural"; I'd say that language and the concept of numbers are about as natural as trees, rocks and and hedgehogs.

Originally Posted By: Lilith

you know that there's plenty of mathematicians who do literally believe that some or all mathematical entities are things that exist in themselves, independent of the physical world or human thought, right

If you mean like Plato, then I would agree that they're "nuts".

[Alorael at Large]

Plato is not a mathematician, and he believed that everything was an entity in itself.

 

—Alorael, who supposes that could explain the phenomenon of all trees drawn by small children having exactly one large knothole. The platonic ideal of tree has just such a knothole. The platonic ideals of specific species of tree, of course, lack them.

[Monroe]

Originally Posted By: Celtic Minstrel

Originally Posted By: Lilith

you know that there's plenty of mathematicians who do literally believe that some or all mathematical entities are things that exist in themselves, independent of the physical world or human thought, right

If you mean like Plato, then I would agree that they're "nuts".

Asking someone to believe in numbers as supernatural entities is like asking someone to believe that humans have a soul. It doesn't seem so terribly far fetched to me.

[Sss-Chah]

Originally Posted By: waterplant

Nouns are products of language. Numbers are products of consciousness - a horse doesn't eat 6 apples. As far as the horse is concerned he or she eats the food that's accessible and most attractive.

Originally Posted By: The Ratt

2


Hey look!!! It's a number!!!

Prove it.

looks like a numeral to me.

[Slawbug]

So basically, Plato was a strictly object-oriented philosopher.

[Lilith]

i'm gonna go cook up a plato food

[waterplant]

Originally Posted By: Lilith

i'm gonna go cook up a plato food

mmm...souvlaki.

[Monroe]

Originally Posted By: CRISIS on INFINITE SLARTIES

So basically, Plato was a strictly object-oriented philosopher.

He believed that every object had a kind of soul. He would say that we call all apples apples because there is a heavenly form of apple upon which all earthly apples are based, and we know of the heavenly apple because this knowledge is inherent in our own souls, which is why such knowledge, indeed all knowledge, seems to come 'naturally' to us. So he wasn't object-oriented, more soul-oriented. Or, as he called them, forms.

[Slawbug]

I was making a joke about object-oriented programming, which is quite Platonic indeed. Classes are ideal forms, while Objects have substance. It even covers the inherent knowledge bit in the form of Class fields and interfaces.

[Monroe]

Heh, clever.

[Erebus the Black]

Originally Posted By: CRISIS on INFINITE SLARTIES

I was making a joke about object-oriented programming, which is quite Platonic indeed. Classes are ideal forms, while Objects have substance. It even covers the inherent knowledge bit in the form of Class fields and interfaces.

OK then, next subject is: Factories and Design-patterns, is programming really better with them?

[waterplant]

Originally Posted By: No_More_PLL_Ever

Originally Posted By: CRISIS on INFINITE SLARTIES

I was making a joke about object-oriented programming, which is quite Platonic indeed. Classes are ideal forms, while Objects have substance. It even covers the inherent knowledge bit in the form of Class fields and interfaces.

OK then, next subject is: Factories and Design-patterns, is programming really better with them?

Better ask one of the guys with a clipboard - I just sweep the floors around here...

[Dintiradan]

I saw some posts on Cantor diagonalization, and got all excited and ready to post. Then I realized that the topic was three pages instead of one. Ah, well.

 

Anyway, if you find the concept of multiple infinities tough, check out this excerpt from Pi in the Sky, which is the most casual and approachable explanation of the technique I've ever seen. If you're a fan of the style, also check out Hotel Infinity, which takes place in an alternate reality where, instead of being mathematicians, Cantor was a hotel owner and Kronecker was a garbage man.

 

(If you're an educator, do take a look at Pi in the Sky. It's a great way to get students interested in math. The Puzzling Adventures of Doctor Ecco is also great, and when I was looking for an Amazon link I discovered that there's a sequel. Joy!)

 

Originally Posted By: Slartucker

I was making a joke about object-oriented programming, which is quite Platonic indeed. Classes are ideal forms, while Objects have substance. It even covers the inherent knowledge bit in the form of Class fields and interfaces.

This analogy always confused me, and makes me question if I really understand Plato. Classes are a combination of interface (specification of the object's qualities) and method implementation (actions that can be performed with the object). Is this how Platonic forms work? What makes it really confusing is how virtual machines for many languages store classes as objects; even the Class class is an object, and the Class class inherits from the Object class (got that?). How does this coincide with reality? Is God just the universal bootloader? I hesitate to say "just"; it's an impressive feat.

 

So. What philosophical schools do other languages follow? Lisp? Ada? Or, dare I ask, LOLCODE, INTERCAT, or Whitespace?

 

Quote:

Factories and Design-patterns, is programming really better with them?

Design patterns in general? Of course. Using them for small projects might be overkill, but at the very least it's good to know where the pitfalls are. Factories? There are good reasons why we have them. They can be a pain to write, especially if you need to implement them as singletons and have to worry about stuff like concurrency. A project I was working on was ported to Spring a year and a half back, and I was introduced to dependency injection. Haven't had to write a factory since.

 

 

 

 

 

Avadon! Wooo!

[Slawbug]

The point of the analogy is that a Platonic ideal form is the idea of a generic object that we hold in our heads. It is the "blueprint" that we compare substantiations of that form to. I think the similarity to objects derived from classes is straightforward.

 

I suppose a procedural programming language would work for Aristotlian philosophy, what with the First Cause and Unmoved Mover and such.

 

Declarative programming languages sounds like a good bet for existentialism ("Whence Come We? What Are We? Whither Go We?").

 

Concatenative languages: analytic philosophy.

Expression-oriented languages: logical positivism.

Non-structured languages: most everything else the Greeks did.

[Celtic Minstrel]

You forgot functional languages. No, those are not the same as procedural languages.

[Slawbug]

Actually, I thought of those too. I hate them. Scheme was not fun for me, and that lack of fun is burned into my mind. But I couldn't settle on a philosophy to pair them with. What do you think?

[Lilith]

Functional languages are a subset of expression-oriented languages anyway.

[Jadan]

Originally Posted By: waterplant

Nouns are products of language. Numbers are products of consciousness - a horse doesn't eat 6 apples. As far as the horse is concerned he or she eats the food that's accessible and most attractive.

On the contrary, I believe a horse is fully capable of realizing if it has eaten 6 apples or 3 apples. Maybe not in so many words, but assuredly his demeanor and attitude towards you would be different if you continued to 6 apples instead of feeding him 3 and keeping 3 for yourself.

[Slawbug]

reacting differently != abstract realization.

 

if I have 3 apples on a paper plate, and I take them off, it might blow away in the wind. it won't do that if I leave the apples on it. that isn't because the paper plate realizes the difference between 3 and 0.

 

now the horse has a nervous system and brain involved in its reactions so obviously it falls somewhere between the extremes of 'paper plate' and 'human'... and for all I know, maybe it does process concepts of number abstractly. however, the fact that it reacts to changes in quantity does nothing whatsoever to support that theory.

[Randomizer]

Originally Posted By: Jadan

On the contrary, I believe a horse is fully capable of realizing if it has eaten 6 apples or 3 apples. Maybe not in so many words, but assuredly his demeanor and attitude towards you would be different if you continued to 6 apples instead of feeding him 3 and keeping 3 for yourself.

That's not abstract reasoning, but horse sense. Food goes to me not you.

[Dantius]

Originally Posted By: Randomizer

Originally Posted By: Jadan

On the contrary, I believe a horse is fully capable of realizing if it has eaten 6 apples or 3 apples. Maybe not in so many words, but assuredly his demeanor and attitude towards you would be different if you continued to 6 apples instead of feeding him 3 and keeping 3 for yourself.

That's not abstract reasoning, but horse sense. Food goes to me not you.

You can lead a horse to advanced abstract mathematics, but you can't make him think?

[Jadan]

Originally Posted By: CRISIS on INFINITE SLARTIES

reacting differently != abstract realization.

You can only assume it is abstract realization. One has a no evidence as to whether the horses reaction is acknowledgement of a quantitative advantage or simply a desire to eat. Reguardless, Eating more or less is still a generic terminology involving mathematics.

[Randomizer]

Originally Posted By: Jadan

Eating more or less is still a generic terminology involving mathematics.

No, it can be simple stimulus and response. If no signal that you are full, then keep eating. It's a simple do loop with at least one conditonal. You could add conditionals for signals outside the body like some one removing the food and or eater.

[everyday847]

Originally Posted By: Jadan

Reguardless, Eating more or less is still a generic terminology involving mathematics.

No, comparisons aren't defined in terms of mathematical relations. Quite the opposite; you can't define a field and hope that its structure will just tell you how to order its elements. (Though because of external intuition because we have concepts of eating more and less or whatever, some orderings seem more natural than others.)

[Jadan]

Originally Posted By: everyday847

No, comparisons aren't defined in terms of mathematical relations. Quite the opposite; you can't define a field and hope that its structure will just tell you how to order its elements. (Though because of external intuition because we have concepts of eating more and less or whatever, some orderings seem more natural than others.)

Comparisons of less than greater than equal to are most definitely mathematical equations themselves. Certainly not something one would suppose to be normal, however. 3x9+8 = (100/4)+10 3x8+8 < (100/4)+10
Equal to Less than Greater than

Originally Posted By: Randomizer

No, it can be simple stimulus and response. If no signal that you are full, then keep eating. It's a simple do loop with at least one conditonal. You could add conditionals for signals outside the body like some one removing the food and or eater.

Well, stimulus and response implies something of a different variety. E.G. A horse that has had enough running, will not run anymore, a child who is too tired, simply will not stay awake.
A Horse that is full however, will continue to eat if given the option, and should a new visitor appear, the horse will change his attention to the new arrival, until finding out whether this new person has any food, or no food.
It's a simple issue known as "You have more than me, and I want it regardless of if I have any or not."

Also, I doubt a horse's brain works much like this

IF(OBJECT)=APPLES
THEN.DEVOUR# #

[Alorael at Large]

Comparisons are by definition not equations. Equations require that equals sign. But your point is otherwise correct.

 

—Alorael, who wouldn't be too sure about horses. He's fairly sure that's how dogs' brains work.

[Student of Trinity]

Horse consciousness is a red herring here. If horses do think about math much as humans do, then horses simply cease to be a useful subject for this discussion. I think we should assume for the purpose of this discussion that horses are pretty dumb, whether or not they actually are.

 

It's an interesting question to me whether or not animals run on algorithms. Specifically, I have often wondered whether the halting problem or Gödel undecidability have any relevance to biology, or for that matter, physics.

 

My guess so far is that, although biology is very complicated, it is not truly algorithmic. Or else it implements an extremely trivial logical system, along the lines of, "what happens, happens". And so in this case there will be no Gödelian point at which a bunch of electrons can't decide what to do next, or an animal runs in a loop forever. I mean, at some point hunger functions as a Turing-proof kill switch, and an animal will give up whatever confusing routine it is running to go and eat.

 

But I am not at all happy with my level of understanding of this. I'm pretty sure I'm very confused about it.

[waterplant]

Originally Posted By: Like Mana from the Pacific

Comparisons are by definition not equations. Equations require that equals sign. But your point is otherwise correct.

—Alorael, who wouldn't be too sure about horses. He's fairly sure that's how dogs' brains work.

My dog gets this *'this dog has not eaten for 5 minutes - please give generously'* look whenever food appears.

[Khoth]

This thread seems sufficiently offtopic for me to bring up a fake proof I quite like:

 

Theorem: A continuious function which passes through the origin is zero everywhere

 

Proof: By strong induction. I'll consider only the x ≥ 0 case, the x < 0 case is done the same way.

Let f(x) be such a function.

 

We know f(0) = 0

 

For the induction hypotheis, suppose f(s) = 0 for 0 ≤ s < x. To complete the induction, I need to prove f(x) = 0. Suppose f(x) = y ≠ 0. But f(x/2) = 0 and f is continuous, so by the intermediate value theorem there is a x/2 < t < x such that f(t) = y/2 ≠ 0.

 

But 0 < t < x so we know f(t) = 0 from the induction hypothesis, a contradiction. So f(x) must be 0. QED.

[Khoth]

I think Gödel's theorems limit thinking ability in approximately the same way that relativity limits running speed.

[Student of Trinity]

My memories of this stuff are pretty old now, but I think it may be quite true that if a continuous function f is zero for a range of s < x, then f(x) = 0. But the fact that f(0) = 0 does not seem to me to imply the initial induction hypothesis, that there exists some x > 0 for which 0 ≤ s<x implies f(s) =0.

 

And establishing f(x) = 0 does not seem to me to establish a 'next step' in the induction hypothesis, i.e., that there is some new x'>x such that 0 ≤ s<x' implies f(s) = 0.

 

In other words, I am skeptical about extending induction to the continuum. And I do find this relevant to the previous irrelevancy, in that part of my skepticism about the relevance of algorithms to physics and biology is that I think the real world is probably continuous, but everything I've ever heard about algorithms seems fundamentally discrete.

 

I like the comparison of indeterminacy to relativity. I rather doubt that the analogy is actually very close, but I think the principle that a fundamental limit may be irrelevant in practice is illustrated well by relativity, and applicable to Gödel.

[Jadan]

Sorry about this..I'm not so numerically eloquent as verbally. But doesn't 0 = 0 when Multiplied or Divided regardless? Rendering the entire proof slightly unneeded? Or does f(x) f(s) and f(t) intend another meaning besides multiplication..

Sorta lost me on that..