• Banno
    8.8k
    "1" has the superficial grammar of a noun, but this is misleading.

    Rather "1" is to be understood through its role in the process of counting. It is understood in learning how to count, not in pointing to individuals.

    And of course this goes for other mathematical entities, too. They are things we do, not things we find.
  • jgill
    723
    And of course this goes for other mathematical entities, too. They are things we do, not things we find.Banno

    Baloney. Are not ideas things we "find?" Whenever we discover a concept, is that not a "find?" Is a Hilbert space something we "do?" So there. :nerd:
  • Banno
    8.8k
    Are not ideas things we "find?"jgill

    No, they aren't.
  • Baden
    10.7k


    Funny you brought up Hilbert, who was a proponent of mathematical structuralism.

    'Hilbert said that in a proper axiomatization of geometry, “one must always be able to say, instead of ‘points, straight lines, and planes’, ‘tables, chairs, and beer mugs’”'

    "Every theory is only a scaffolding or schema of concepts together with their necessary relations to one another, and ... the basic elements can be thought of in any way one likes."

    https://www.iep.utm.edu/m-struct/

    @Banno You editing a wiki article again?
  • Pneumenon
    443
    Counting this way is correct, because we make it so. And there are cultures that say, "One, two, many," and don't go past a certain number.

    But suppose there were a culture that said, "2 + 2 = 5," and their definitions of two, five, equality, and addition were the same as ours. If they all agree, they can't be wrong, can they?
  • Baden
    10.7k


    Why would they do that?
  • ZzzoneiroCosm
    985
    But suppose there were a culture that said, "2 + 2 = 5," and their definitions of two, five, equality, and addition were the same as ours. If they all agree, they can't be wrong, can they?Pneumenon

    Not wrong. Insane.
  • Baden
    10.7k
    The point here (I presume) is that numbers and other mathematical entities are placeholders in systems that were originally set up because they were of cultural value. Stuff needed to be done. The system helped people to do it. The elements of the system arose from that process.
  • frank
    5.1k
    "Santa Clause" appears to be a noun, but this is misleading.

    Rather, "Santa Clause" is to be understood through its role in separating people from their money near the winter solstice.

    This goes for other entities as well. Superman, for instance, is a thing we do, not a thing we find.
  • jgill
    723
    Are not ideas things we "find?" — jgill


    No, they aren't.
    Banno

    I open a math book and find a new definition. Is that not a thing I find?

    Metal detectors find buried coins. I suppose that is the naive notion you entertain. :roll:
  • Banno
    8.8k
    If set the task of describing the motion of drops of water on a window pane, we might well make use of 1+1=1.

  • Banno
    8.8k
    I open a math book and find a new definition. Is that not a thing I find?jgill

    Some one else put it there.
  • ZzzoneiroCosm
    985
    The raindrop in reverse: 1-1=2.
    The split worm: 1=2.
  • Banno
    8.8k
    ...naive...jgill

    Maybe. It comes from Wittgenstein. Do you think him naive?

    This thread is a branch from @Sam26's Summary of the Tractatus.
  • StreetlightX
    5.9k
    Go big or go home Banno - nothing refers to anything, not even words! - unless used in that way, of course.

    It's as they say about guns: words and numbers don't refer, people do.
  • Banno
    8.8k
    You editing a wiki article again?Baden

    Not this time. Trying to read Wittgenstein’s Philosophy of Mathematics but I'm stuck on the way Rodych uses intension and extension.

    So the extension of a set is the actual items in the set. The intension of the set is the rule by which one decides what is included - the property had my the members of the set. Rodych says that extensions must be finite.

    Why?

    One cannot physically list the integers. But in understanding the intension of "integer" we understand how to construct the extension... and in so doing it seems to me that we understand the extension to be infinite.
  • Banno
    8.8k
    Yeah, but "StreetlightX" refers to StreetlightX in a way that "1" does not refer to 1.

    That is, whole there (presumably) is a StreetlightX, there is not a 1.
  • Banno
    8.8k
    But suppose there were a culture that said, "2 + 2 = 5," and their definitions of two, five, equality, and addition were the same as ours. If they all agree, they can't be wrong, can they?Pneumenon

    That's not what I'm after here. What you have constructed is a contradiction; they can't both claim to be adding in the way we do and that 2+2=5; we simply apply radical interpretation to work out which of the terms in "2+2=5" they are using differently, and change our interpretation to match theirs.
  • jgill
    723
    I open a math book and find a new definition. Is that not a thing I find? — jgill

    Some one else put it there.
    Banno

    Tell me clearly what the word "find" means.

    MW: "to discover by the intellect or the feelings" or "to come upon by searching or effort " or . . .

    This thread is a good example of why philosophy appears sometimes to be "garbage in = garbage out"

    When you start with a really shoddy definition things go downhill quickly. IMHO

    :chin:

    Maybe. It comes from Wittgenstein. Do you think him naive?Banno

    I made a small attempt to read him years ago but found little connection with the world of mathematics In which I lived.
  • Banno
    8.8k
    Yes, that. And mathematical Platonism is wrong.
  • Banno
    8.8k
    So you are well placed to help me.

    Would you be able to take a look at 2.2 Wittgenstein’s Intermediate Finitism?

    Can you follow the discussion of extension?
  • StreetlightX
    5.9k
    Yeah, but "StreetlightX" refers to StreetlightX in a way that "1" does not refer to 1.Banno

    I suppose, but only because it's used that way right now.

    I guess - to be less facetious - I don't want math to be anything all that special. It's a language too. The distinction in use between "StreetlightX" and "1" is intra-linguistic, and not between language and some other, special script.

    But otherwise yes, "1" obviously doesn't refer to anything at all.
  • ZzzoneiroCosm
    985
    StreetlightX" refers to StreetlightX, whereas "1" refers to "1".
  • Banno
    8.8k
    I don't want math to be anything all that special.StreetlightX

    I've sympathy for that. I think it more like doing proper grammar than like metal detecting.
  • Banno
    8.8k
    "1" refers to "1"ZzzoneiroCosm

    No, it doesn't. It doesn't refer to anything.
  • jgill
    723
    "Mathematics as Human Invention: According to the middle Wittgenstein, we invent mathematics, from which it follows that mathematics and so-called mathematical objects do not exist independently of our inventions. Whatever is mathematical is fundamentally a product of human activity."

    More or less. We invent, but we also discover. Creating vs discovering is a topic of interest occasionally for math people. Once we have invented we have set in play a process of unraveling or discovering what logically follows. Along the way we invent again, and follow paths stemming from those activities.

    "it follows that 'the mathematical infinite' resides only in recursive rules"

    As an analyst, I agree. I am not prone to use the infinity symbol like any other.

    I'll read more later and report back. :cool:
  • Banno
    8.8k
    From The Stanford article:
    Given that we have invented only mathematical extensions (e.g., symbols, finite sets, finite sequences, propositions, axioms)...
    Why finite sets? This seems an idiomatic use; and I;m not sure if it comes from Wittgenstein or Rodych

    An infinite mathematical extension (i.e., a completed, infinite mathematical extension) is a contradiction-in-terms

    Well, yes - if one assumes that extensions are finite, then... extensions are indeed finite.
  • Banno
    8.8k
    Cheers.

    And my apologies for baiting you with the OP. I needed a mathematician who might disagree with a constructivist approach to mathematics; the ruse worked.
  • Pneumenon
    443
    Why would they do that?Baden

    Why not?

    What you have constructed is a contradiction; they can't both claim to be adding in the way we do and that 2+2=5Banno

    Same as my question to Baden. Why do we assume that radical translation must yield the same thing we have?

    And to both of you: I'm not just being difficult. It's all quite relevant, I think.
  • Banno
    8.8k
    Why do we assume that radical translation must yield the same thing we have?Pneumenon

    Charity.
  • Wayfarer
    9.9k
    They are things we do, not things we find.Banno

    My dog does one’s and two’s. The one’s I can generally ignore, but the two’s I have to clean up.
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