Mathematical Theory (controversy!)

  • Richard Warr (6/6/2011)


    toddasd (6/6/2011)


    [quote-0I think Michael Valentine nailed it with the "Skepticism in education" post. My friend asked me the same thing at lunch on Friday, "Can you give me 0.999... of a penny? " I handed him a penny. He didn't get it.[/quote-0]

    That's where we differ. I would have taken a penny, shaved an infinitessimally small piece off it, and given him that 😉

    And that's the basis of the logically valid but mathematically untrue proof that all numbers are equal to all other numbers. All you have to do is multiply (or divide) both sides of an equation by infinity or zero, after proving infinity = zero. And that proof is trivial.

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  • GSquared (6/6/2011)


    mtillman-921105 (6/6/2011)


    toddasd, then how do you resolve one of Zeno's paradoxes? Basically (and mathematically), if one runner is catching up with another by half the distance per minute, then how long will it take for the runner to catch up with the other? According to strict math, the runner would never catch up. But we know that it isn't true from a practacle standpoint. The difference becomes so small that it makes no discernable difference. There are other paradoxes like that too by the way.

    There's an infinity of difference between "...so small that it makes no discernable difference. ..." and "zero difference". Unless we assume space has an absolute minimum value, in which case, once the fraction of difference is less than half of that value, it will be forced to become zero difference, because of rounding. But that's an unnecessary assumption.

    Again, the problem is assuming "mathematics" has some sort of idependent existence in some theoretical "objective universe". Math is a language. It is arbitrary, as languages must be. There is no math in whatever "objective reality" is. It's a mental construct, and it's terms are based on postulated definitions.

    Assuming Zeno's paradox applies to this also assumes that 1 != 1 for any value of 1. If you can't see that, then you need to simply study up on the postulated definitions of numbers. (I've already gone over this in prior posts in this thread to the extent that I care to for this purpose. Further study is warranted if my explanation isn't complete enough for you.)

    GSquared, I'll even go one further. A true statement has to be both valid and sound. Mathematics can only be valid or not valid, but can not be sound. Therefore, mathematics can be said to be neither true nor false. (I think that last part is actually a quote too, But I have no time to look it up right now.)

    [Edited for GSquared's quote]

    The greatest enemy of knowledge is not ignorance, it is the illusion of knowledge. - Stephen Hawking

  • mister.magoo (6/6/2011)


    (Breaking my own promise to keep out - sorry 😛 )

    This thread is RBAR... I have a set based solution:

    http://www.ocf.berkeley.edu/~wwu/cgi-bin/yabb/YaBB.cgi?board=riddles_medium;action=display;num=1027804564

    No tally tables!

    haha, love the SQL reference.

    and got a good laugh out of that thread too. Right away they start with

    A non-math solution:

    What number is between .9999... and 1?

    See? There isn't one. Hence:

    .9999... = 1

    Of course there is, we just can't express it with the Arabic numeral that we use. Lets create a new digit # which is bigger than 9 but less than 10.

    Then .999999... < .####..... < 1 so there is a number between .999... and 1, but we just can't express it with the digits that we NORMALLY use. Just because we can't express it doesn't mean that it isn't there.

    Let's just create a new numbering system out of thin air to satisfy the dissent.

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  • mtillman-921105 (6/6/2011)


    Richard Warr (6/6/2011)


    toddasd (6/6/2011)


    [quote-0I think Michael Valentine nailed it with the "Skepticism in education" post. My friend asked me the same thing at lunch on Friday, "Can you give me 0.999... of a penny? " I handed him a penny. He didn't get it.[/quote-0]

    That's where we differ. I would have taken a penny, shaved an infinitessimally small piece off it, and given him that 😉

    GSquared, I'll even go one further. A true statement has to be both valid and sound. Mathematics can only be valid or not valid, but can not be sound. Therefore, mathematics can be said to be neither true nor false. (I think that last part is actually a quote too, But I have no time to look it up right now.)

    (I think you may have quoted the wrong post, but I think I know where you wanted to come from on this.)

    The problem with that statement about true/false is, again, it assumes a binary and absolute true vs false, which doesn't bear out in "the real world".

    For example, it's true enough for most people that "there are 7 billion people in the world", but others will need more true statements, like "there are 6.7 billion people in the world", or even more true "there are 6.7 billion (US, not UK) people in the world". ("Billion" means two different things on different sides of The Pond.) Each of those is "more true" than the prior. (The actual numbers are immaterial to the point here, but should be "true enough".)

    1 + 1 = 2 is true. 1 apple + 1 apple = 2 apples is true. But the degree of truth is unverified. 2 > 1 is true. 2 apples > 1 apple may or may not be true, depending on what value we are measuring in the apples. For example, if your context is "quantity of food", 1 really big apple might be greater than 2 small ones. So, truth is relative and contextual as well as precise/imprecise.

    Without context, math can be completely true, so long as it is completely based on mutually known and understood postulates. But it will never be absolutely true in anything other than a purely mathematical context. But neither is anything else.

    - Gus "GSquared", RSVP, OODA, MAP, NMVP, FAQ, SAT, SQL, DNA, RNA, UOI, IOU, AM, PM, AD, BC, BCE, USA, UN, CF, ROFL, LOL, ETC
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  • GSquared (6/6/2011)..."Billion" means two different things on different sides of The Pond.

    wow. I was about to question this when I looked it up, totally threw me for a loop.

    How can

    1 billion apples = 1,000,000,000 apples in the US

    1 billion apples = 1,000,000,000,000 apples in the UK

    How is this possible! We're not even dealing with infinity here, it's a definite finite amount! Is this just a difference of meaning in the same word?

    dictionary link!

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  • calvo (6/6/2011)


    GSquared (6/6/2011)..."Billion" means two different things on different sides of The Pond.

    wow. I was about to question this when I looked it up, totally threw me for a loop.

    How can

    1 billion apples = 1,000,000,000 apples in the US

    1 billion apples = 1,000,000,000,000 apples in the UK

    How is this possible! We're not even dealing with infinity here, it's a definite finite amount! Is this just a difference of meaning in the same word?

    dictionary link!

    *blinks* What.


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  • Come on Mr. Credentials! Help us understand!

    (Mr. Credentials = GSquared. The previous statement is true)

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  • calvo (6/6/2011)


    GSquared (6/6/2011)..."Billion" means two different things on different sides of The Pond.

    wow. I was about to question this when I looked it up, totally threw me for a loop.

    How can

    1 billion apples = 1,000,000,000 apples in the US

    1 billion apples = 1,000,000,000,000 apples in the UK

    How is this possible! We're not even dealing with infinity here, it's a definite finite amount! Is this just a difference of meaning in the same word?

    dictionary link!

    Yep.

    Everyone say it with me, "Math is a language. Nothing more, nothing less."

    English != American. Just ask Professor Higgins.

    - Gus "GSquared", RSVP, OODA, MAP, NMVP, FAQ, SAT, SQL, DNA, RNA, UOI, IOU, AM, PM, AD, BC, BCE, USA, UN, CF, ROFL, LOL, ETC
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  • calvo (6/6/2011)


    Come on Mr. Credentials! Help us understand!

    (Mr. Credentials = GSquared. The previous statement is true)

    Or at least true enough. 🙂

    - Gus "GSquared", RSVP, OODA, MAP, NMVP, FAQ, SAT, SQL, DNA, RNA, UOI, IOU, AM, PM, AD, BC, BCE, USA, UN, CF, ROFL, LOL, ETC
    Property of The Thread

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  • GSquared (6/6/2011)


    (I think you may have quoted the wrong post, but I think I know where you wanted to come from on this.)

    The problem with that statement about true/false is, again, it assumes a binary and absolute true vs false, which doesn't bear out in "the real world".

    For example, it's true enough for most people that "there are 7 billion people in the world", but others will need more true statements, like "there are 6.7 billion people in the world", or even more true "there are 6.7 billion (US, not UK) people in the world". ("Billion" means two different things on different sides of The Pond.) Each of those is "more true" than the prior. (The actual numbers are immaterial to the point here, but should be "true enough".)

    1 + 1 = 2 is true. 1 apple + 1 apple = 2 apples is true. But the degree of truth is unverified. 2 > 1 is true. 2 apples > 1 apple may or may not be true, depending on what value we are measuring in the apples. For example, if your context is "quantity of food", 1 really big apple might be greater than 2 small ones. So, truth is relative and contextual as well as precise/imprecise.

    Without context, math can be completely true, so long as it is completely based on mutually known and understood postulates. But it will never be absolutely true in anything other than a purely mathematical context. But neither is anything else.

    Let me try this from another angle, here is how Einstein put the idea:

    “As far as the laws of mathematics refer to reality, they are not certain; and as far as they are certain, they do not refer to reality.”

    This also applies to logic itself by the way.

    The greatest enemy of knowledge is not ignorance, it is the illusion of knowledge. - Stephen Hawking

  • mtillman-921105 (6/6/2011)

    “As far as the laws of mathematics refer to reality, they are not certain; and as far as they are certain, they do not refer to reality.”

    This also applies to logic itself by the way.

    "As far as the laws of logic refer to reality, they are not certain; and as far as they are certain, they do not refer to reality."

    Is this what you mean by also applying to logic?

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  • calvo (6/6/2011)


    mtillman-921105 (6/6/2011)

    “As far as the laws of mathematics refer to reality, they are not certain; and as far as they are certain, they do not refer to reality.”

    This also applies to logic itself by the way.

    "As far as the laws of logic refer to reality, they are not certain; and as far as they are certain, they do not refer to reality."

    Is this what you mean by also applying to logic?

    From a formal philosophical standpoint, yes. As I said, for a statement to be true or false (in logic), it has to be both valid and sound. But only validity applies in logic (and math) itself.

    An illustration might help. 1 + 1 = 2 right? Well, what if we're talking about putting together one atom and one anti-matter atom (anti-matter is in the news today by the way)? In that case, isn't 1 + 1 = 0 since the anti-matter annihilates the matter and nothing is left? So what do we conclude from this example, that math is wrong? No, the math is still valid, but mis-applied (not sound).

    The greatest enemy of knowledge is not ignorance, it is the illusion of knowledge. - Stephen Hawking

  • yeah, antimatter contained and observed for 16 minutes and 40 seconds! how cool is that!

    Perhaps a bad example 🙂

    Along GSquare's line of thinking, just the fact that you are adding "things" (atoms in this case) makes this a more philosophical argument than pure mathematical. The number 1 added to the number 1 will always have a sum of 2. Now only because you've associated this formula to how atoms react when introduced (outside the realm of pure mathematics) you must create a whole new language, as GSquared would say. A language where

    h=hydrogen ah=antihydrogen

    1h + 1ah = 0.

    1ah + 1ah = 2

    But I get the point. The philisophical argument behind all this is excellent. The concept of infinity is stunning. The idea that decimals are an expression for sums is great.

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  • mtillman-921105 (6/6/2011)


    GSquared (6/6/2011)


    (I think you may have quoted the wrong post, but I think I know where you wanted to come from on this.)

    The problem with that statement about true/false is, again, it assumes a binary and absolute true vs false, which doesn't bear out in "the real world".

    For example, it's true enough for most people that "there are 7 billion people in the world", but others will need more true statements, like "there are 6.7 billion people in the world", or even more true "there are 6.7 billion (US, not UK) people in the world". ("Billion" means two different things on different sides of The Pond.) Each of those is "more true" than the prior. (The actual numbers are immaterial to the point here, but should be "true enough".)

    1 + 1 = 2 is true. 1 apple + 1 apple = 2 apples is true. But the degree of truth is unverified. 2 > 1 is true. 2 apples > 1 apple may or may not be true, depending on what value we are measuring in the apples. For example, if your context is "quantity of food", 1 really big apple might be greater than 2 small ones. So, truth is relative and contextual as well as precise/imprecise.

    Without context, math can be completely true, so long as it is completely based on mutually known and understood postulates. But it will never be absolutely true in anything other than a purely mathematical context. But neither is anything else.

    Let me try this from another angle, here is how Einstein put the idea:

    “As far as the laws of mathematics refer to reality, they are not certain; and as far as they are certain, they do not refer to reality.”

    This also applies to logic itself by the way.

    We're saying the same thing. Different words, different details, same concept.

    - Gus "GSquared", RSVP, OODA, MAP, NMVP, FAQ, SAT, SQL, DNA, RNA, UOI, IOU, AM, PM, AD, BC, BCE, USA, UN, CF, ROFL, LOL, ETC
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  • mtillman-921105 (6/6/2011)


    calvo (6/6/2011)


    mtillman-921105 (6/6/2011)

    “As far as the laws of mathematics refer to reality, they are not certain; and as far as they are certain, they do not refer to reality.”

    This also applies to logic itself by the way.

    "As far as the laws of logic refer to reality, they are not certain; and as far as they are certain, they do not refer to reality."

    Is this what you mean by also applying to logic?

    From a formal philosophical standpoint, yes. As I said, for a statement to be true or false (in logic), it has to be both valid and sound. But only validity applies in logic (and math) itself.

    An illustration might help. 1 + 1 = 2 right? Well, what if we're talking about putting together one atom and one anti-matter atom (anti-matter is in the news today by the way)? In that case, isn't 1 + 1 = 0 since the anti-matter annihilates the matter and nothing is left? So what do we conclude from this example, that math is wrong? No, the math is still valid, but mis-applied (not sound).

    Actually, that's kind of what negative numbers are for. 1 + -1 = 0.

    - Gus "GSquared", RSVP, OODA, MAP, NMVP, FAQ, SAT, SQL, DNA, RNA, UOI, IOU, AM, PM, AD, BC, BCE, USA, UN, CF, ROFL, LOL, ETC
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