Quick question:

Am I wrong to state that words, composed of letters, define concepts - concrete and abstract, while numbers represent a certain relationship between concepts?

mindhack wrote:Quick question:

Am I wrong to state that words, composed of letters, define concepts - concrete and abstract, while numbers represent a certain relationship between concepts?

Probably an oversimplification, but not exactly wrong as such. Numbers define concepts that are abstract, but which can have concrete instances. A number also defines the relationship that is extant between the entities which, when grouped together appropriately, form a concrete instance of the abstract concept of the particular number in question. I hope that doesn't sound too much like metaphysical wibble!

Calilasseia wrote:

mindhack wrote:Quick question:

Am I wrong to state that words, composed of letters, define concepts - concrete and abstract, while numbers represent a certain relationship between concepts?

Probably an oversimplification, but not exactly wrong as such. Numbers define concepts that are abstract, but which can have concrete instances. A number also defines the relationship that is extant between the entities which, when grouped together appropriately, form a concrete instance of the abstract concept of the particular number in question. I hope that doesn't sound too much like metaphysical wibble!

Let's see if I follow.

Numbers define abstract concepts, such as relationships between concepts. These abstract concepts, represented by numbers, may represent concrete instances.

For example when relevent data is determined and put into a proven formula a specific abstract concept(?) is made concrete.

More or less. What you're doing, when you do this, is producing a specific instance of the class. If those numbers connect to a real world situation in some way, you're making a categorical statement about the relevant concrete entities, whose behaviour is governed by that equation, and in particular, stating that those entities will enjoy a specific relationship when the relevant numbers are substituted into the variables.

One of the reasons mathematics requires a certain finesse on the part of its educators, is the verbosity that can result when applying rigour.

it's interesting. I dare to say I do follow you, but that my (written) english isn't adequate.

Basically, I think the core of what you so elequantly explain should be taught in lesson 1 to all kids starting with mathematics.

It makes the subject tangible.

Thank you very much!

Calilasseia wrote:Put three apples on the table in front of you. You now have a concrete instantiation of the abstract concept known as 'three'.

Put three oranges on the table in a neat row in front of the apples. You now have another concrete instantiation of that abstract concept.

You don´t. You have in one case a number 3, and in the other case an amount (i.e. a physical quantity here not expressed in it´s SI unit). My table has a cetain lenght, the number that goes with the unit is meaningless without the unit (measured in tables it´s 1, measured in cm, inches, lightyears...). The SI unit here would be moles - you have 4.98*10-2 moles of apples there.

Calilasseia wrote:This means that the only true instances of ed(Q,x) are ed(Q,1) and ed(Q,Q).

But this is the definition of a prime number, as we started with above. Which means that Q is a prime number.

Q isn´t neccessarily prime - it may stil have prime divisors larger than Qmax (also leading to the proof).


But as we noted at the start, Q > pmax, our hypothetical 'maximum' prime number. Therefore pmax is not the maximum prime number, which contradicts our initial hypothesis. This applies whatever value we choose for our proposed pmax - we can always construct a larger prime number than pmax, as shown above.

Therefore the initial postulate, that there exists a maximum prime number, is false. 30031 is the first one for which this holds (1*2*3*5*7*11*13+1), which is 59*509

To the OP:

Seattledru wrote:if numbers have neither a beginning nor an ending then how can one say they truly exist, are they not merely concept and as such cannot "control" any real theory, now letters on the other hand are finite they have a beginning and an ending and as such would be more realistically used when contemplating theories? Sorry f this sounds choppy but its a new thing I am trying to grasp and decided today to get others perspective on it.

a) It doesn´t matter if numbers exist or not - the key question is whether the statements we can make about them are true. A notion of truth that requires existence runs into an infinite regress and leads to a position where nothing can be known.
b) Scientific theories themselves are abstract entities - they are sets of predicted observations and quite usually infinite sets of predicted observations. Mathematics allows us to handle sets and infinite ones in particular rather well.
c) Letters aren´t quivalent to numbers, but to digits. A string of letters is a base 26 representation of a number. As such, you end up with the same thing, just written in a format less familiar to people (this post as a string of ASCII symbols is also a number).

susu.exp wrote:

Calilasseia wrote:This means that the only true instances of ed(Q,x) are ed(Q,1) and ed(Q,Q).

But this is the definition of a prime number, as we started with above. Which means that Q is a prime number.

Q isn´t neccessarily prime - it may stil have prime divisors larger than Qmax (also leading to the proof).

But the proof begins with the postulate that there are no larger prime numbers than pmax. Therefore all we need to demonstrate is that there IS at least one larger prime than pmax, without relying on any a priori assumption to this effect, in order to complete the proof, which is achieved by the proof as I presented it. That proof does not assume in advance that there are primes larger than pmax, it simply takes that postulate as given, then uses the primes up to and including pmax, which, according to the terms of the postulate, are the only primes we can rely on without assuming a priori that larger primes exist. Using those primes, the proof demonstrates that a prime larger than pmax exists, and can be constructed regardless of whatever value pmax is asserted to be. QED.

I got put off number/set theory when someone told me that "3" was the set of all sets with 3 elements.

3 = { {{{}},{}},{{}},{} } i think

Calilasseia wrote:[But the proof begins with the postulate that there are no larger prime numbers than pmax. Therefore all we need to demonstrate is that there IS at least one larger prime than pmax, without relying on any a priori assumption to this effect, in order to complete the proof, which is achieved by the proof as I presented it. That proof does not assume in advance that there are primes larger than pmax, it simply takes that postulate as given, then uses the primes up to and including pmax, which, according to the terms of the postulate, are the only primes we can rely on without assuming a priori that larger primes exist. Using those primes, the proof demonstrates that a prime larger than pmax exists, and can be constructed regardless of whatever value pmax is asserted to be. QED.

I think you misunderstood me. My point was that the proof does not in fact construct a larger prime that pmax, it merely shows that one exists, because Q is not divisible by any prime up to pmax. It is a non-constructive proof. If Qi=1+Product (j=1..i) pj where pj is the jth prime was prime, computing lots of large primes would be a trivial problem. We know that there are infinitely many primes, but constructing the set of primes is not trivial at all.

As a counterexample that Q isn't necessarily a prime:

3*3+1 = 10

10 = 2*5

5>3

This popped into my mind earlier today ; )

susu.exp wrote:

Calilasseia wrote:[But the proof begins with the postulate that there are no larger prime numbers than pmax. Therefore all we need to demonstrate is that there IS at least one larger prime than pmax, without relying on any a priori assumption to this effect, in order to complete the proof, which is achieved by the proof as I presented it. That proof does not assume in advance that there are primes larger than pmax, it simply takes that postulate as given, then uses the primes up to and including pmax, which, according to the terms of the postulate, are the only primes we can rely on without assuming a priori that larger primes exist. Using those primes, the proof demonstrates that a prime larger than pmax exists, and can be constructed regardless of whatever value pmax is asserted to be. QED.

I think you misunderstood me. My point was that the proof does not in fact construct a larger prime that pmax, it merely shows that one exists, because Q is not divisible by any prime up to pmax. It is a non-constructive proof. If Qi=1+Product (j=1..i) pj where pj is the jth prime was prime, computing lots of large primes would be a trivial problem. We know that there are infinitely many primes, but constructing the set of primes is not trivial at all.

Ah, but Q is relatively prime to all of the primes asserted to exist in the postulate. Since that postulate asserts that the only primes that exist are those up to and including pmax, demonstrating that Q is prime with respect to all of those primes, is sufficient to establish that the list is incomplete, and does not require us to assume the conclusion in advance. Which is why I presented the proof in that form as being a rigorous one, for it erects no additional assertions, and relies only upon the terms of the original postulate, in order to falsify that postulate.

home_ wrote:As a counterexample that Q isn't necessarily a prime:

3*3+1 = 10

10 = 2*5

5>3

This popped into my mind earlier today ; )

Actually, you should have used (2×3)+1 in order to be consistent with the terms of my proof. Because if you assert that pmax=3, then Q = (2×3)+1 = 7, which is not divisible by either 2 or 3, and therefore demonstrates that any list of primes containing only containing 2 and 3 is incomplete. Likewise, if you assert that pmax=5, then in this instance, you have that Q = (2×3×5)+1 = 31. 31 is not divisible by 2, 3 or 5. Therefore this list of three primes is incomplete. My proof, as presented above, simply demonstrates that this is true in all possible cases, for any proposed finite list, and therefore that said finite list is incomplete.

If we look at various values for Q, depending upon the choice of pmax, we end up with the following list:

pmax=2 : Q = 3
pmax=3 : Q = 7
pmax=5 : Q = 31
pmax=7 : Q = 211
pmax=11 : Q = 2,311
pmax=13 : Q = 30,031
pmax=17 : Q = 510,511
pmax=19 : Q = 9,699,691
pmax=23 : Q = 223,092,871
pmax=29 : Q = 6,469,693,231

If you go here, you can check (as I did) if each of the numbers above is prime. In the above list of values of Q, we have:

Q=30,031 is divisible by 59, but 59 is not part of the asserted "complete" list that generated it.
Q=510,511 is divisible by 19, but 19 is not part of the asserted "complete" list that generated it.
Q=9,699,691 is divisible by 347, but 347 is not part of the asserted "complete" list that generated it.
Q=223,092,871 is divisible by 317, but 317 is not part of the asserted "complete" list that generated it.
Q=6,469,693,231 is divisible by 331, but 331 is not part of the asserted "complete" list that generated it.

It would be interesting to see if any higher values of Q are prime, though sadly my calculator quickly reaches its limits, with respect to maximum integer representation, and once it moves into floating point, it cannot be relied upon to produce exact integer divisions once the magnitude of the number exceeds a certain point.

Calilasseia wrote:Actually, you should have used (2×3)+1 in order to be consistent with the terms of my proof.

I already knew (a version of) this proof before and I didn't bother to carefully read yours. Sorry, and you're right , I wasn't consistent with your version. But it also works if you allow any power of those primes, it doesn't change anything.

Calilasseia wrote:It would be interesting to see if any higher values of Q are prime, though sadly my calculator quickly reaches its limits, with respect to maximum integer representation, and once it moves into floating point, it cannot be relied upon to produce exact integer divisions once the magnitude of the number exceeds a certain point.

It's not hard to overcome this obstacle: you can just write a program (C or Java or whatever you like) that stores digits in an array (or even in a separate file) instead of 'regular integer', and then write your own operation of division that works with array elements. You get whatever magnitude of precision you like and the only obstacle is amount of your computer memory (or hard drive space, if you just write digits in a file) and a bit longer waiting time to get result. ; )

Anyway, thank you for posting those few interesting numbers.

PARI/GP would work as well. (I´m currently trying to get to grips with how to import data to it, because I need to perform some bignum calculations on data and while I´ve got an extension for R that allows rationals, aparently it runs into Rs lenght limit for strings).

And yes, if the proof merely states that Q was relatively prime or coprime to all primes up to pmax it´d be completely correct. But maths and nitpicking are to a great degree the same thing...

Or a modern high-level language (say Python), most of which are sensible enough to support bignums out of the box and use them by default, only providing unboxed fixed-width integers when you really need their efficiency.

Seattledru wrote:

[Reveal] Spoiler: previous conversation...

The_Metatron wrote:

Seattledru wrote:if numbers have neither a beginning nor an ending then how can one say they truly exist, are they not merely concept and as such cannot "control" any real theory, now letters on the other hand are finite they have a beginning and an ending and as such would be more realistically used when contemplating theories? Sorry f this sounds choppy but its a new thing I am trying to grasp and decided today to get others perspective on it.

I'm going to have a go at this, too.

First, the topic title:

"Are mathmatical theroies legit?"

To which I would answer: More legitimate than the words "mathmatical", and "theroies", both misspelled.

Secondly, you are conflating the characters used to represent words with the words themselves when you say letters are finite. Indeed, the letters of an alphabet are finite, as are the digits we use to represent numbers. But, you are freely allowing any number to exist, while denying an infinite combination of words with a similarly finite character set to represent them as the Arabic numerals are used to represent any number.

For example, there have been some finite number of words written since the dawn of writing and the moment you read this post. Yet, here's a perfectly new sequence that until this moment, never existed: "The rhino chicken soldered the copper pipes in his bathroom to his nose with an ice cream scoop."

Words are not finite, because of the combinations they can be used. No different from any number. Just a particular combination of digits.

Misspellings are a good way to see someones approach to a subject, if they point them out then that shows certain arrogance that pretty much say 'ignore whatever else they say" rather to see a newbies interest in the subject themselves, then to ignore the fact I was focused on letters, not words the alphabet a-z 26 letters no more no less rather than words which we all know to be infinite, Geez.

No. I'm calling bullshit on this.

I don't fail to note how you conveniently discount the concept of words, and latch onto the 26 letters of the Roman alphabet, calling them finite. You even more conveniently say nothing about the plain fact that numbers are expressed nicely using the smaller set of ten Arabic numerals, but still enjoy infinity.

Count the space between words as a character, which it most certainly is, and you lose the distinction between letters and words.

Your manufactured dichotomy does not exist. IthinkyouknowperfectlywellwhatI'mtalkingabout.

As for your little spelling test, I call bullshit on that, too. Try to be so cavalier with symbology in mathematics and see how far you get with it. In this regard, mathematics doesn't enjoy the same built in error correction that written language does. Everyone knew what ideas you intended to communicate even with your crap spelling. Mathematics is not so forgiving. The "eyeglasses" on the Hubble Space Telescope are a monument to that. Its triboal to denomfrate teh biolt-im errur correjtoim usinf wurds.

Indeed, with respect to the above on error correction in spoken language, there's a comedy sketch from [i]The Two Ronnies[i] that is pertinent here, namely:

I have two comments on the OP, one neat, one messy:
1. Numbers don't control anything. They are used in various laws of nature, but these are Descriptive Laws, not Prescriptive Laws. They don't give orders to the particles and waves, they are just useful for describing what said particles and waves are likely to do next.

2. Wondering about whether numbers exist is almost certainly a dead end. Indeed, I am starting to wonder whether the concept of existence is too elusive of definition, for an assertion of the existence or non-existence of anything to have any meaning. But I need to mull over my muddled thoughts on that rather longer before posting on it directly.