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quote:
Originally posted by MBM:
quote:
Originally posted by HonestBrother:

I specialize in Number Theory.


From a universal perspective, aren't numbers really pretty much concepts without meaning?


No more than 'justice', 'freedom', or 'beauty'.

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Since numbers are infinite, isn't the difference between any two numbers meaningless?


It depends upon the point of view that one adopts. I'll return to this question when I have more time and my head is clear.

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If numbers are supposed to attach value and describe difference, then in the realm of theory when we consider all numbers, how do we differentiate one from another? It seems like their values are all "fungible" if there are an infinite number of numbers on either side of them on a number line.


There are a number of ways one can classify numbers and make real distinctions between them. For example, some whole numbers are prime numbers - like 2, 3, 5, 7, 11, 13, 17, 19, etc. - while other numbers are composite - like 4, 6, 8, 9, 10, 12, 14, etc.

The numbers in the first group can be expressed as a product of two whole numbers in one and only one way: 5=1x5 and that's it. 13=1x13 and that's it.

But 6=1x6 and 2x3. 18=1x18, 2x9, and 3x6. So both 6 and 18 are composite and not prime.

So here we've discover a real (and important) difference between these two groups of numbers.

I thought it was interesting that in the movie 'Contact', the extraterrestrials chose to transmit the sequence of primes as their first contact message with another intelligent species.

Number Theory studies such properties in depth.

There are any number of further statements one can make along these lines.

For example, it is very easy to see that the sequence of prime numbers is an infinite list... and so is the sequence of composites....

So interestingly we've decomposed the infinite list of whole numbers into two different infinite sublists.

We can go even further. For example, one can show that there are infinitely many prime numbers that leave a remainder of 1 when divided by 4: like 5, 13, 17, 29, 37, 41, etc.

But there are also infinitely many primes that leave a remainderof 3 when divided by 4: 3, 7, 11, 19, 23, 31, 43, etc.

Notice that these lists are different and that aside from 2, these two infinite lists exhaust the infinite list of primes. Every prime will appear in one and exactly one of these lists.

Furthermore, you can differentiate between two numbers based solely on their divisibilty properties.

You can play these games forever. We've only scratched the surface in terms of looking at ways to differentiate between numbers.

Far from being trivial, many throughout the ages have asserted that God is, in fact, the ultimate mathematician.

I'll come back to this topic.
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quote:
Originally posted by HonestBrother:

No more than 'justice', 'freedom', or 'beauty'.


But perhaps that's just the issue. The concepts that you mention are personal, relative, and subjective ones. Numbers and mathematics have always been "advertised" as being a part of something that is decidedly not those things. In reality - they may be no less "soft" than something like "beauty" - clearly they are at least as nebulous. I guess this point is pushed even further when we think about the impact that quantum physics has on this. 1 + 1 is not always 2 after all! 17

quote:

There are a number of ways one can classify numbers and make real distinctions between them. For example, some whole numbers are prime numbers - like 2, 3, 5, 7, 11, 13, 17, 19, etc. - while other numbers are composite - like 4, 6, 8, 9, 10, 12, 14, etc.


I guess this begs the difference of the meaning or import of these distinctions. Again, as you note, if there are an infinite number of these prime and composite numbers - I could see how someone would argue that this difference isn't even as significant as the difference between different grains of sand on the Earth?

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We've only scratched the surface in terms of looking at ways to differentiate between numbers.


I hear you, but again, from a universal perspective - the difference between any two numbers is infinite. Additionally, on a number line, there are an infinite number of points between any two numbers on that line. Hence - it seems that if this concept of numbers (as we understand it) is valid, then numbers and math are just as fluid and soft and relative as any other man-made concept - like love or niceness or - as you mention - justice. No?

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Far from being trivial, many throughout the ages have asserted that God is, in fact, the ultimate mathematician.


Perhaps - but if God started counting as fast as He could (and boy, that's fast! 16 ) at the very begining of the universe until now - He still wouldn't have even gone even one point further on the Universal Number Line. 9 God may be the ultimate mathematician - but so what? What does that mean? Since God is the ultimate message board designer or gangsta rapper or cribbage player or pencil maker - is there real meaning or significance to this point in terms of us better understanding the meaning of numbers or math? Confused 15

P.S. My grandfather's PhD was in math so I had similar conversations with him growing up. It makes me smile to discuss this again with you HB! 15
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MBM:
I guess this begs the difference of the meaning or import of these distinctions. Again, as you note, if there are an infinite number of these prime and composite numbers - I could see how someone would argue that this difference isn't even as significant as the difference between different grains of sand on the Earth?


Let me be sure I'm understanding your point... Are you saying that just because a particular category may have a very large number of members - or even infinitely many members - that there is therefore no difference between individual members of the category? Smile .... Wouldn't you say there was a difference between yourself and your child regardless of the population of the planet?... No matter that there are 4 million people on earth or 4 billion or even 4 trillion, aren't you different from your child?

Yes.. there are infinitely many primes but we can still make meaningful distinctions between them.

The phrase "an even prime number" points to a unique number: the number 2. There is only one number that answers to this description. The phrase "an odd prime which divides every number which ends in 0" names the number 5 and no other number. We can give a ton of further examples ...

By the way, could you so easily distinguish yourself from every other human being on the planet? You know, for example, that you're not the only Michael Moore... You might say that no one else has your DNA... But could you articulate exactly what makes your DNA different from everybody else's? ... Smile

And yes there are infinitely many primes that leave a remainder of 1 when divided by 4 AND ALSO infinitely many primes which leave a remainder of 3 when divided by 4... BUT these two classifications make a clear distinction between the numbers 41 and 43 - which also happen to be twin primes by the way - but that's another subject ... Smile

The phrase "the black guy" might not distinguish you from every other human being on the planet. But it might be enough to distinguish you from everyone in your place of employment. Just so, the phrase "a prime which leaves a remainder of 1 when divided by 4" is not enough to distinguish 41 from 5 or 13 or 17. But it is enough to distinguish it from 43. If we wish, we could further distinguish 41 from 5 or 13 by adding that it also leaves a remainder of 1 when you divide it by 8 ... Smile

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Additionally, on a number line, there are an infinite number of points between any two numbers on that line. Hence - it seems that if this concept of numbers (as we understand it) is valid, then numbers and math are just as fluid and soft and relative as any other man-made concept - like love or niceness or - as you mention - justice. No?


You think like a Topologist: that branch of mathematics which studies the qualitative (as opposed to quantitative) aspects of space. From this point of view there's no difference between a doughnut and a coffee cup. If you imagine that these objects were made of rubber or some other flexible material, then you could distort one object into the other without doing too much violence (like ripping or tearing). Just think of the doughnut hole as corresponding to the hole between the cup and handle.

From this perspective, there's no difference between any two open-ended intervals on the number line. So your observations are good from this perspective.

But again, it depends on the point of view you choose to take. From one point of view, there are qualities that you have, MBM, that make you just like other people (your race or your name for example). But there are also qualities you have which make you unlike anyone else... Smile

The number 2 is like infinitely other numbers in being even. It's unlike any other in being an even prime.

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Far from being trivial, many throughout the ages have asserted that God is, in fact, the ultimate mathematician.


God may be the ultimate mathematician - but so what? What does that mean? Since God is the ultimate message board designer or gangsta rapper or cribbage player or pencil maker - is there real meaning or significance to this point in terms of us better understanding the meaning of numbers or math? Confused 15



It makes more sense if one understands mathematics not as the science of number but rather as the science of order. Mathematicians don't study numbers. We study patterns. We study order, structure, and relationship in their purest manifestations. (A mathematical equation is in essence an assertion of relationship - numerical identity - between the entities on either side of the equal sign).

If you've seen the movie "A Beautiful Mind" this point is made in a dramatic way. John Nash is portrayed as someone with a gift for seeing patterns in the world. This is probably the best portrayal of a mathematician I've ever seen in popular media. Most other things depict us as weird people who are obsessed with numbers... we're just fancy accountants some people think... No... It's not about numbers .... It's about patterns and relationships between them....

So God is the ultimate mathematician in the sense that God is the architect of the universe. The ultimate arbiter of ORDER and STRUCTURE.

I'm reading a book called "The Elegant Universe" by the physicist Brian Greene. In it, he makes the following statement:

"... the universe would be a vastly different place if the properties of the matter and force particles were even moderately changed. For example, the existence of stable nuclei forming the hundred or so elements of the periodic table hinges delicately on the ratio between the strengths of the strong and electromagnetic forces. The protons crammed together in atomic nuclei all repel one another electromagnetically; the strong force among their constituent quarks, thankfully, overcomes this repulsion and tethers the protons tightly together. But a rather small change in the relative strengths of these two forces would easily disrupt the balance between them, and would cause most atomic nuclei to disintegrate. Furthermore, were the mass of the electron a few times greater than it is, electrons and protons would tend to combine to form neutrons, gobbling up the nuclei of hydrogen (the simplest element in the cosmos, with a nucleus containing a single proton) and, again, disrupting the production of more complex elements. Stars rely upon fusion between stable nuclei and would not form with such alterations to fundamental physics. The strength of the gravitational force also plays a formative role. The crushing density of matter in a star's central core powers its nuclear furnace and underlies the resulting blaze of starlight. If the strength of the gravitational force b]were increased[/b], the stellar clump would bind more strongly, causing a significant increase in the rate of nuclear reactions. But just as a brilliant flare exhausts its fuel much faster than a slow-burning candle, an increase in the nuclear reaction rate would cause stars like the sun to burn out far more quickly, having a devastating effect on the formation of life as we know it. On the other hand, were the strength of the gravitational force significantly decreased, matter would not clump together at all, thereby preventing the formation of stars and galaxies.

We could go on, but the idea is clear: the universe is the way it is because the matter and force particles have the properties they do [corresponding to specific numbers like mass] ...."

I'm not a physicist but I've always been fascinated by Physics and the fact that the universe itself seems to obey mathematical equations... Starting with the simple observation that objects moving in a gravitational field follow trajectories (i.e., orbits) which are conic sections (Kepler) - ellipses, parabolas, hyperbolas, and circles. They have a precise mathematical description. Einstein's Theory of Relativity, at it's core, is a handful of equations that make extremely powerful (and accurate) predictions about objects moving through space and time.

So God - the Designer of the Universe - is a mathematician in this sense. Mathematics is the language of Physics. It is apparently also the language of God.

So far, I'm not aware of gangsta rap making a difference to understanding reality beyond this planet... Big Grin

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P.S. My grandfather's PhD was in math so I had similar conversations with him growing up. It makes me smile to discuss this again with you HB! 15


Very good... Smile

Maybe if I'm fortunate enough I'll have these conversations with my children and grandchildren one day ...

I'd love one day to be able to take my kid out with a telescope and marvel together at the night time sky ... Smile
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quote:
Originally posted by HonestBrother:

Are you saying that just because a particular category may have a very large number of members - or even infinitely many members - that there is therefore no difference between individual members of the category?


Yeah - I guess. 15 I'm expressing my frustration at the discordance between thinking about numbers in the traditional sense - that they define precise and discrete quantities, versus in the universal sense in that they are infinite and relatively difficult to define relative to other numbers. Quantum physics tells us that 4 does not always equal 4 - sometimes it equals 3 or 5. So in an absolute sense - numbers are relative and transitory. Furthermore, in a relative sense, numbers are even more difficult to define it would seem. Again, on a number line it is almost impossible to discern between numbers since there are an infinite amount of numbers on either side of the number and in between any two numbers - whether between 1 and 3 or 1 and a googol.

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Wouldn't you say there was a difference between yourself and your child regardless of the population of the planet?


Yes, but neither I nor humanity are in infinite supply.

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The phrase "an even prime number" points to a unique number: the number 2.


Right - and this is a meaningful distinction - until you juxtapose quantum physics against this where that two might not always equal two. Confused

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The phrase "the black guy" might not distinguish you from every other human being on the planet. But it might be enough to distinguish you from everyone in your place of employment.


And because there are a finite number of guys and black guys makes that distinction meaningful - contrasted with the number of numbers - which is infinite.


quote:
You think like a Topologist: that branch of mathematics which studies the qualitative (as opposed to quantitative) aspects of space. From this point of view there's no difference between a doughnut and a coffee cup. If you imagine that these objects were made of rubber or some other flexible material, then you could distort one object into the other without doing too much violence (like ripping or tearing). Just think of the doughnut hole as corresponding to the hole between the cup and handle.


I think that went over my head, but - how would you respond to my confusion about numbers on a number line: what is the significance of differences between numbers when there are an infinite number of points between any two numbers and on either side positive and negative?
quote:
Originally posted by MBM:
how would you respond to my confusion about numbers on a number line: what is the significance of differences between numbers when there are an infinite number of points between any two numbers and on either side positive and negative?


Back... Smile

Let's put to the side for one moment the question of quantum logic. This only unnecessarily confuses things.

So the question is how do we distinguish between two things within a set of infinite size?

But I will answer your question with a question: How do we distinguish between any 2 things no matter how many others there are of that type?

If it were impossible to make distinctions, it would be impossible to make the claim that there are infinitely many such things: if numbers were truly indistinguishable they couldn't be counted at all. Right? If any one number were as good as any other?

Here's an example.

Think of the span of time between 1:00 pm yesterday and 1:00 pm today. Let's think of this time span as a line connecting those two points in time.

1:00 pm yesterday]_______________________________[1:00 pm today

Between any two moments in that time span there will always be other moments in time. There are in fact infinitely many such moments. From your experience, are these moments indistinguishable from each other?

From my own experience, I can definitely tell the difference between moments where say I'm making love and moments where I'm teaching class...
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quote:
Originally posted by HonestBrother:
quote:
Originally posted by MBM:
how would you respond to my confusion about numbers on a number line: what is the significance of differences between numbers when there are an infinite number of points between any two numbers and on either side positive and negative?


Back... Smile

Let's put to the side for one moment the question of quantum logic. This only unnecessarily confuses things.

So the question is how do we distinguish between two things within a set of infinite size?

But I will answer your question with a question: How do we distinguish between any 2 things no matter how many others there are of that type?

If it were impossible to make distinctions, it would be impossible to make the claim that there are infinitely many such things: if numbers were truly indistinguishable they couldn't be counted at all. Right? If any one number were as good as any other?

Here's an example.

Think of the span of time between 1:00 pm yesterday and 1:00 pm today. Let's think of this time span as a line connecting those two points in time.

1:00 pm yesterday]_______________________________[1:00 pm today

Between any two moments in that time span there will always be other moments in time. There are in fact infinitely many such moments. From your experience, are these moments indistinguishable from each other?

From my own experience, I can definitely tell the difference between moments where say I'm making love and moments where I'm teaching class...


This discussion is probably best held while listening to Pat Methany or Weather Report and passing a bong. Big Grin

But it would seem {taking a deep hit} that the difference between numbers and, say, moments in time would be {blowing out several smoke rings, followed by a thin blueish cloud of smoke} that numbers are purely conceptual; whereas, moments have characteristics that are observable and reportable. {Passing the bong to the right}

"Hey, the pizza's not here, yet??? Well, some much for 30 minutes or less."
quote:
Originally posted by Kweli4Real:
quote:
Originally posted by HonestBrother:
quote:
Originally posted by MBM:
how would you respond to my confusion about numbers on a number line: what is the significance of differences between numbers when there are an infinite number of points between any two numbers and on either side positive and negative?


Back... Smile

Let's put to the side for one moment the question of quantum logic. This only unnecessarily confuses things.

So the question is how do we distinguish between two things within a set of infinite size?

But I will answer your question with a question: How do we distinguish between any 2 things no matter how many others there are of that type?

If it were impossible to make distinctions, it would be impossible to make the claim that there are infinitely many such things: if numbers were truly indistinguishable they couldn't be counted at all. Right? If any one number were as good as any other?

Here's an example.

Think of the span of time between 1:00 pm yesterday and 1:00 pm today. Let's think of this time span as a line connecting those two points in time.

1:00 pm yesterday]_______________________________[1:00 pm today

Between any two moments in that time span there will always be other moments in time. There are in fact infinitely many such moments. From your experience, are these moments indistinguishable from each other?

From my own experience, I can definitely tell the difference between moments where say I'm making love and moments where I'm teaching class...


This discussion is probably best held while listening to Pat Methany or Weather Report and passing a bong. Big Grin

But it would seem {taking a deep hit} that the difference between numbers and, say, moments in time would be {blowing out several smoke rings, followed by a thin blueish cloud of smoke} that numbers are purely conceptual; whereas, moments have characteristics that are observable and reportable. {Passing the bong to the right}

"Hey, the pizza's not here, yet??? Well, some much for 30 minutes or less."



Somebody say "bong"?

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Originally posted by virtue:
quote:
Originally posted by HonestBrother:
Kresge, how exactly do you say 'Kresge'?



Good question....

I've been wondering this myself....

Kresge? I've been pronouncing it

Krezz' gee (accent on the first syllable)...

Is that correct?


Peace,
Virtue

Virtue,
That is the correct pronunciation. Folks from the midwest would probably recognize it as the original name for K-mart. The K stands for Kresge.
ma'am: Although I should have known that when the great one known to mere mortals as ma'am chose to grant us a visual depiction of herself, she would choose a golden frog, is there a special reason?

Also

DDR: Dance Dance Revolution?

INTP: Introverted iNtuitive Thinking Perceiving?

But W-Indy-American? Please don't tell me you're an Independent Bush voter ...

And IC3?

Smile
quote:
Originally posted by HonestBrother:
ma'am: Although I should have known that when the great one known to mere mortals as ma'am chose to grant us a visual depiction of herself, she would choose a golden frog, is there a special reason?

Also

DDR: Dance Dance Revolution?

INTP: Introverted iNtuitive Thinking Perceiving?

But W-Indy-American? Please don't tell me you're an Independent Bush voter ...

And IC3?

Smile


Hehe. The frog is random, but kinda cute.

Noooo Bush love from me.

West Indian-American.

IC3 is English po-po code for a suspect Black.
quote:
Originally posted by EbonyRose:
I had no idea this thing was 10 pages!! Eek So, if this has been asked & answered, I apologize! Smile


Honest Brother ... what made you decide to become a teacher? Confused



Your confusion confuses me ....

You don't think I'd be any good at it? Smile


Really it's part of the job. I wanted to be a mathematician. If you work at a university, teaching is a part of the job.

It starts off in grad school: As part of your assistantship you have to teach lower level classes. Either that or assist a professor in some way (grading for example).

Unfortunately, many in my profession regard teaching as a necessary evil they endure to do the part of the job they really like: research.

But over the years, I've come to enjoy it.

It's nothing like teaching grade school or high school though. Nothing at all.

You're not anywhere near as much a substitute parent. Nowhere near as responsible for your students.

It doesn't require quite the same skills and personal qualities.
quote:
Originally posted by HonestBrother:
Your confusion confuses me ....

You don't think I'd be any good at it? Smile


No, actually, I guess I was wondering just what you answered ... why you would pick to teach rather than do research or something else in your field of choice. I never knew anybody that answered the question "What do you want to be when you grow up?" by saying "A mathemetician"! Big Grin

Though my parents wouldn't let me use a calculator, I thought it was the best invention since white bread! lol

Big tfro to you for your choice, though!
ma'am...
I guess my interest was kickstarted in several different ways, travelling through the West Indies a few years back, and finding it difficult to find any contemporary novels there - especially by female authors - someone I was dating and thought I would marry, and my general curiosity about hot climate cultures - being a hot climate gurl myself.

At this point in time, in between my set reading list of African American non-fiction and history books, I've been enjoying some female authors and their perspectives. I'm trying to read through some of the 'heavyweight' Black writers first, as well as the 'independants' - who are writing on blogs, music lyrics, poetry and spoken word rather than books and essays.

As well as hearing names online and through friends, I picked up a book of essays from "minorities" around the globe and was mightily impressed by author Zora Neale Hurston in particular. ...and yes, Toni Morrison, Maya Angelou, lol. Wink I also really like Bell Hooks - she is so on point and I've gained a clearer picture of contemporary issues - male and female - through her. And E Dandicat's books. And so many more yet to read...

I finished reading Hurston's 'Their Eyes Were Watching God' a few weeks back and absolutely loved the book - it's language, honesty, human frailty, insight and unfailing sense of feminine mystique and sensuality. Parts of that book gave me goosebumps and I was moved to want to start a thread and I may still do that. Smile
.
quote:
Originally posted by FireFly:
ma'am...
I guess my interest was kickstarted in several different ways, travelling through the West Indies a few years back, and finding it difficult to find any contemporary novels there - especially by female authors - someone I was dating and thought I would marry, and my general curiosity about hot climate cultures - being a hot climate gurl myself.

At this point in time, in between my set reading list of African American non-fiction and history books, I've been enjoying some female authors and their perspectives. I'm trying to read through some of the 'heavyweight' Black writers first, as well as the 'independants' - who are writing on blogs, music lyrics, poetry and spoken word rather than books and essays.

As well as hearing names online and through friends, I picked up a book of essays from "minorities" around the globe and was mightily impressed by author Zora Neale Hurston in particular. ...and yes, Toni Morrison, Maya Angelou, lol. Wink I also really like Bell Hooks - she is so on point and I've gained a clearer picture of contemporary issues - male and female - through her. And E Dandicat's books. And so many more yet to read...

I finished reading Hurston's 'Their Eyes Were Watching God' a few weeks back and absolutely loved the book - it's language, honesty, human frailty, insight and unfailing sense of feminine mystique and sensuality. Parts of that book gave me goosebumps and I was moved to want to start a thread and I may still do that. Smile
.


I've never heard of Dandicat.

I don't think I've really read any novels by Caribbean women. In my life. Besides children's books, short stories, and prose. Nothing.

I just noticed that.

*notes Dandicat*

Are there any Aussies we should add to our reading lists over here.
quote:
Originally posted by ma'am:

I've never heard of Dandicat.

I don't think I've really read any novels by Caribbean women. In my life. Besides children's books, short stories, and prose. Nothing.

I just noticed that.

*notes Dandicat*
FYI it's Danticat and her first name is Edwidge... She is a Haitian born author... I have only read two of her works "Krik? Krak!" and "Breath, Eyes, Memory" I highly recommend her...

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