Regarding genes and the environment ... Does the environment include intangibles?

[Boojatta]

Second question: Can you describe a non-empty set of positive whole numbers that has no least element?

The obvious answer to the second question is "no", but that answer can be amplified. Not only are you unable to describe such a set, but no such set exists. We are confident that no such set exists, but what is our confidence based on?

Is it based primarily on the cultural environment, which includes textbooks that assert that no such set exists, and teachers who punish unorthodox answers to the second question with low marks?

Is it based primarily on human genes, so that such a set might exist, but people would have to struggle against their instincts to accept that it exists?

Is our confidence based primarily on past observations, making it potentially like a false conjecture that seems too difficult to prove, but for which a counter-example is eventually discovered? In other words, is it not something that should be a foundation for mathematical conclusions, but is itself in as much need of justification as any theorem that one might wish to prove?

[Teaser]

1. Our confidence is based on the definition of positive whole numbers

.

[Boojatta]

5. People were confident of the second statement before ...

... anybody had invented a way of defining the positive whole numbers.

The original message of this thread can be rewritten by replacing the second statement with something else. Perhaps you can suggest an alternative to the second statement that would avoid this tangent?

Certainly theorems cannot be based on nothing but definitions. Definitions merely provide a way to rewrite statements to replace defined symbols with undefined (aka "primitive") symbols. After rewriting a statement so that it contains no defined symbols, it needs to be somehow justified, either by recognizing that it should be accepted as an axiom, or by deriving it from axioms.

[Teaser]

6. Were they confident of it?

I don't know. I don't know if they even thought about it.

[BadgerKid]

7. Not all whole numbers are positive. n/t

[Teaser]

8. but all positive whole numbers are positive

.

[BadgerKid]

9. Indeed,

[mike_c]

2. unless I'm mistaken, the following set of positive whole numbers has no least element....

{5,5,5,5,5}

More generally, let x be any positive whole number. Any set y composed of any number of repetitions of x will have no least element.

on edit: I suspect you meant a set of ORDERED positive whole numbers?

[Boojatta]

3. {5,5,5,5,5} = {5}

The least element is 5, which is also the only element.

[mike_c]

4. that must be why I'm not a mathematician....

That means that there is only one positive whole number 5 in the entire universe, i.e. all 5's are the same. I do see that-- five is always five and never anything else. And your original problem did cite the set of positive whole numbers, not sets of positive integer quantities (I presume you'll agree that 5 oranges != 5 apples). So yes, I do agree with you. I just didn't see the problem from that perspective, which is why I'll likely never make a good mathematician or a logician! :rofl:

[Orrex]

10. In college I dated a woman who was really good at gelato

One thing that we had in common was our shared annoyance at overblown language used to mask empty rhetoric.

[Boojatta]

11. How do you determine whether or not some language is overblown?

The word "rhetoric" suggests that there is something illegitimate but simply applying the adjective "rhetoric" without any explanation of the basis for doing so is little more than name-calling.

[Boojatta]

12. Let us proceed on the assumption that what sparked your contribution

to this thread was its content and not the username Boojatta.

I encourage you to ask questions and make comments. For example, do you think that you can guess what idea I had in mind when I wrote a particular part of the Original Post of this thread, a particular part that you would like to draw attention to, but think that I failed to express the idea very clearly? Can you describe at least two different ideas where only one idea would be indicated if I had chosen my words and organized my sentences more carefully?

Is there some particular part of the Original Post of this thread, a particular part that you would like to draw attention to, where I either asserted or hinted at some claim that you believe to be false?

Did I invoke or apply an unstated principle that remains excessively vague and that I should explicitly formulate?

Above are merely some suggestions for continuing the promising new path of discussion that has been opened up by your contribution. Please don't allow them to restrict your next contribution. I am posting them not to narrow the discussion, but to show that there are at least a few different paths available. Be bold. Strike out on a new path and shine light where it hasn't yet shone.

[dimbear]

13. The simplest set of positive whole numbers with no least element

would probably be the exceptions to the Goldbach conjecture. These are all the even numbers > 2 and not expressible as the sum of two primes. I assert that there is no smallest element of this set.

If you have any quibble with my statement, please provide a counterexample.

[Boojatta]

14. That's undeniably clever.

Interestingly, one thing you rely upon is a reverse onus.

I assert that there is no smallest element of this set.

What is the basis for your assertion? For example, what train of thought led you to suspect that your assertion is true? Have you gone beyond suspicion and constructed an argument in support of the assertion?

If you have any quibble with my statement, please provide a counterexample.

I have resolved to refrain from quibbling. What if I have no quibble, but instead I have doubt? Keep in mind that while Goldbach's conjecture continues to be neither proven nor disproven, I am at liberty to doubt Goldbach's conjecture and to also doubt the negation of Goldbach's conjecture.

[tama]

21. Riemann hypothesis and anthropic principle

Provability of Goldbach's conjecture about prime numbers is related to the provability Riemann Hypothesis and its relation to distribution of prime numbers. Let us first suppose or accept, without quibble, that RH is not provable in any finite set of axioms, but gödelian "unprovable truth". Riemann hypothesis has close connection with quantum theory (http://en.wikipedia.org/wiki/Hilbert%E2%80%93P%C3%B3lya_conjecture#Possible_connection_with_quantum_mechanics), and hence central or we could say, axiomatic, to the way world with beings like us manifests. In this sense, with the additional presupposition of platonical realism, RH equals anthropic principle, it can be hypothesized that quantum mechanics and all the physics that follow from it including us and human mathematical cognition are dependent from what RH states about distribution of prime numbers.

Simple test about that hypothesis is falsification by human mathematical cognition that succeeds in creating or finding distribution of primes dissimilar to RH.

It should be also noted that such a test of RH could be similar to the test suggested for proving Many-Worlds hypothesis, namely playing Russian Roulette. If the test subject does not succeed blowing his brains out but continues to play some possible world, then the test subject has proven MWH at least to herhimself, but not necessarily to friends and relatives who mourn him in other possible worlds. Needless to say, ethics of such tests are at least as questionable as testing if atom bomb ignites the atmosphere or does not, as calculations predicted.

[Boojatta]

15. Kick to encourage people to answer some of the questions

[Name removed]

16. Deleted message

Message removed by moderator. Click here to review the message board rules.

[Boojatta]

17. Kick

[Boojatta]

18. Kick

[Boojatta]

19. Kick

[tama]

20. In philosophy of math

there are two ontological views that don't fail in carefull scrutiny, namely platonical realism (http://plato.stanford.edu/entries/platonism-mathematics/) and fictionalism (http://plato.stanford.edu/entries/fictionalism-mathematics/). Semantically they are equivalent, but fictionalistic narrative tells other story about the ontological status of World of Forms ("Platonia") that platonical realism accepts as given. Given substratum of and beyond any and all notions about space and time, which in that sense, is more real than fenomenalistic physical world in 4D-spacetime or any other mathematical form.

Hence, it would seem that in platonistic ontology, actualizations and implications of DNA would manifest the possibilities allowed by Platonia, including mathematical cognition in humans, in our case quite likely according to some anthropic principle limiting the world of possible worlds to such that allow mathematical cognitions like us. I'm less certain if fictionalistic narrative semantically but not ontologically equivalent to platonical realism would allow a story being told that human mathematical cognition is caused by and limited to DNA genetic code, but judging from the OP, such narrative can at least be presented as a hypothesis.