• Hillary
    1.9k
    Then why does Wolfram say
    ...they are some quantity that is explicitly nonzero and yet smaller in absolute value than any real quantity.
    Banno

    Because infinitesimals are paradoxical.
  • Banno
    25.3k
    ...so the paradox is that Wolfram mathworld has a different definition to you.
  • Agent Smith
    9.5k
    I like this (a lot). So look here, infinitesimals are zeroish, but not zero! You're geniusish but no, you're not a genius! It's a cheat code in the game! You fool the system with ishyness. :snicker:
  • Banno
    25.3k
    No. Infinitesimals are "explicitly not zero".
  • Hillary
    1.9k
    so the paradox is that Wolfram mathworld has a different definition to youBanno

    Is that a paradox? Don't think so. There has never been a mathematical object as paradoxically as the infinitesimal.
  • Agent Smith
    9.5k
    No. Infinitesimals are "explicitly not zero".Banno

    Ok, but they could be described as zeroish. That's the point, oui monsieur?
  • Tate
    1.4k
    Rather, adding certain non-logical axioms results in contradictionsTonesInDeepFreeze


    Could you give an example of a non-logical axiom and explain what makes it non-logical?
  • Banno
    25.3k
    zeroishAgent Smith

    No.

    Is that a paradox?Hillary

    No, that's irony.

    Still not seeing a paradox.
  • Hillary
    1.9k
    No, that's irony.

    Still not seeing a paradox
    Banno



    The paradox lies in the irony. :lol:

    No, seriously how can you create a non-zero interval out of length pieces that have zero length? By taking infinitely many? How you do that? You might counter they don't have zero length and it merely approaches zero, but that begs the question. If the distance between two points decreases, the distance will get zero and they touch. The paradox is that they never touch while able to break on through.
  • Banno
    25.3k
    ...how can you create a non-zero interval out of length pieces that have zero length?Hillary

    But infinitesimals do not have zero length. So that's not what is happening.
  • Hillary
    1.9k
    ...how can you create a non-zero interval out of length pieces that have zero length?
    — Hillary

    But infinitesimals do not have zero length
    Banno

    I wrote:
    You might counter they don't have zero length and it merely approaches zero, but that begs the questionHillary
  • Agent Smith
    9.5k
    No.Banno

    Why?

    How would you round off 0.0001?
  • Banno
    25.3k
    Infinitesimals are nonzero. Think that was mentioned. That's not begging the question, that's called a definition.

    How would you round off 0.0001?Agent Smith

    Life's too short for this. 0.0001 is not an infinitesimal.
  • Hillary
    1.9k
    that's called a definitionBanno

    But the very definition is paradoxical..
  • Agent Smith
    9.5k
    Life's too short for this. 0.0001 is not an infinitesimal.Banno

    Answer the question instead of beating around the bush.

    If 0.0001 rounds off to 0, a fortiori an infinitesimal rounds off to 0, oui? Zeroish.
  • Banno
    25.3k
    But the very definition is paradoxical..Hillary

    How, exactly?
  • Banno
    25.3k
    Answer the question instead of beating around the bush.Agent Smith

    Your question makes no sense.
  • Hillary
    1.9k
    How, exactly?Banno

    Approaching without being able to touch.
  • Banno
    25.3k
    Still not seeing it.
  • Hillary
    1.9k
    Answer the question instead of beating around the bush.Agent Smith

    :lol:

    Exactly, brother Agent!
  • Hillary
    1.9k
    Still not seeing it.Banno

    Consider two points. There is space between them. For the infinitesimal to exist the can never touch. However close they get.
  • Banno
    25.3k
    I have a basic understanding of Continuity. Get on with it. Where's the paradox? Where's the (p & ~ p)?
  • ssu
    8.7k
    For those (like me) who aren't mathematicians, a great way to understand this is to look at the history of mathematics and how much great minds have pondered these question throughout the centuries. Then you get more understanding of how the debate has gone as there actually is a historical narrative how humans have thought about these issues and how we have gotten to where we are.

    The thing is that a mathematicians or math education can (and usually does) look at this ahistorically. They just give you the end result, at worst basically an algorithm to use with not much debate about the underlying issues. A proof is given and that's all. Once you understands let's say the notion of limits, for some it looks quite meaningless to ponder about the motion paradoxes of Zeno. These are basically foundational questions about mathematics, and then mathematicians (or philosophers) do understand the question better from that viewpoint.
  • Hillary
    1.9k
    I have a basic understanding of the continuum. Get on with it. Where's the paradox?Banno

    The paradox lies in the break-up of the continuum in points. If you get on with it a little harder you might see it. The distance between two points gets smaller and smaller. But there will always remain space between them. You can say it's defined like that, but it's non-coherent.
  • Banno
    25.3k
    The SEP article on continuity and infinitesimals is set out in an historical order. It gets complicated. Ah dinnae ken. There might be some stuff in it relevant to the topic, but it's not obvious.
  • Banno
    25.3k
    The paradox lies in the break-up of the continuum in points. If you get on with it a little harder you might see it. The distance between two points gets smaller and smaller. But there will always remain space between them. You can say it's defined like that, but it's non-coherent.Hillary

    There's nothing paradoxical in that. It's just odd, not paradoxical. A topic for investigation, not @Agent Smith's broken maths or broken logic.
  • Hillary
    1.9k
    The very notion of laying infinite infinitely small intervals together is nonsensical. Like the other way round, zero infinite big intervals.
  • Hillary
    1.9k
    There's nothing paradoxical in that.Banno

    A paradox goes against established opinion.
  • Banno
    25.3k
    The very notion of laying infinite infinitely small intervals together is nonsensical.Hillary

    Why?

    The maths works. What more is there?

    A paradox goes against established opinion.Hillary

    But infinitesimals are established opinion.
  • Hillary
    1.9k
    But infinitesimals are established opinion.Banno

    Yes, but the majority of people think the opposite. That establishes the paradox. The twin paradox goes against established opinion (no time delay). Differentials idem dito.
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