Skip to main content
Log In | Register

TR Memescape


Topic: infinities and the mind-dependence of maths (split from Newton etc.) (Read 790 times) previous topic - next topic - Topic derived from Newton, Copernicus, G...

0 Members and 1 Guest are viewing this topic.
  • Brother Daniel
  • Global Moderator
  • predisposed to antagonism
  • 605

  • 160

Re: infinities and the mind-dependence of maths (split from Newton etc.)
Reply #100
So you're happy to invoke a "god's eye view" with respect to the proposition "Bro D made a mathematical error at 22:37 ADT on 8th September 2016 CE" but not with respect to the proposition "2+3=5"?
Not at all. The subjective view is all we have.
But earlier you said "Detecting an error is an objective matter".  What was that about?
Quote from: Testy
but there is no way to know if we have unseen error.
"No way to know" doesn't affect my point at all.  You're implying that unseen error is something that can be there (objectively), even if we don't happen to know about it.  It's the god's eye view again.  You don't seem to be able to avoid appealing to objective truths, even within your "the subjective view is all we have" account.

  • Brother Daniel
  • Global Moderator
  • predisposed to antagonism
  • 605

  • 160

Re: infinities and the mind-dependence of maths (split from Newton etc.)
Reply #101
So we seem to be at an impasse, regarding this Platonist(?) versus constructivist(?) dispute (if that's what we are).

Do you want to have another go at explaining what your objection is to the standard mathematical account of cardinality?  I can't make sense of anything you said on that topic, even if I try to take your constructivist(?) position (if that's what it is) for the sake of argument.

  • el jefe
  • Needs a Life
  • asleep till 2020 or 2024
  • 2,314

  • 479

Re: infinities and the mind-dependence of maths (split from Newton etc.)
Reply #102
Quote from: Testy
The paradox that such a set is neither normal nor abnormal is only a paradox of sleight of hand regarding timing. It is identical to asking if a person is alive or dead when you remove time from the equation.
What.  The.  Hell.  Are.  You.  Smoking.
he's saying it's like schroedinger's cat.  it's like all the same thing, man.  it's all connected.  the whole thing is just an atom in some big alien's fingernail.  and tbh, I'm not digging your negative energy.

  • 7,305

  • 1092

Re: infinities and the mind-dependence of maths (split from Newton etc.)
Reply #103
So you're happy to invoke a "god's eye view" with respect to the proposition "Bro D made a mathematical error at 22:37 ADT on 8th September 2016 CE" but not with respect to the proposition "2+3=5"?
Not at all. The subjective view is all we have.
But earlier you said "Detecting an error is an objective matter".  What was that about?
Quote from: Testy
but there is no way to know if we have unseen error.
"No way to know" doesn't affect my point at all.  You're implying that unseen error is something that can be there (objectively), even if we don't happen to know about it.  It's the god's eye view again.  You don't seem to be able to avoid appealing to objective truths, even within your "the subjective view is all we have" account.
language is not well suited to set the universe in a multi subjective experiential schema.

Anything we can observe is considered objective however each observation is subjective. I'm not really convinced that subjective/objective is a definite distinction. We can agree on experience, rules, lots of things. We can witness a digital readout and agree on the number. We can agree whether or not it is raining, although there is some, er, gray area between rain and very high humidity. I am not describing solipsism. But what is 'objective is the perception. What is subjective is the understanding/cataloging/classifying etc. Modeling/thinking/understanding/whatever you want to call that landscape is the source of subjectivity, and sensory perception the source of objectivity. And yet even those have overlap.

But we can communicate our experiences to some degree and systems of logic are particularly easy to communicate because there are clear boundaries. Anyway, you are taking this to its logical edges which is fine but seems like a lot to do to defend the idea that math exists independently of mind and that infinity can be confidently stated as anything more than a continuum that doesn't have an obvious break.
Love is like a magic penny
 if you hold it tight you won't have any
if you give it away you'll have so many
they'll be rolling all over the floor

  • 7,305

  • 1092

Re: infinities and the mind-dependence of maths (split from Newton etc.)
Reply #104
Quote from: Testy
The paradox that such a set is neither normal nor abnormal is only a paradox of sleight of hand regarding timing. It is identical to asking if a person is alive or dead when you remove time from the equation.
What.  The.  Hell.  Are.  You.  Smoking.
he's saying it's like schroedinger's cat.  it's like all the same thing, man.  it's all connected.  and tbh, I'm not digging your negative energy.
Thanks for the good vibes jefe.
Love is like a magic penny
 if you hold it tight you won't have any
if you give it away you'll have so many
they'll be rolling all over the floor

  • Brother Daniel
  • Global Moderator
  • predisposed to antagonism
  • 605

  • 160

Re: infinities and the mind-dependence of maths (split from Newton etc.)
Reply #105
but seems like a lot to do
no u.  :)

I'm just going with the flow, man.

You're the one who's fighting against language:
Quote from: Testy
language is not well suited
Of course, that doesn't mean I'm right.  But it puts your "seems like a lot to do" comment in rather amusing perspective.

  • el jefe
  • Needs a Life
  • asleep till 2020 or 2024
  • 2,314

  • 479

Re: infinities and the mind-dependence of maths (split from Newton etc.)
Reply #106
...
Quote from: Testy
I have never really gotten the nuances of mathematical descriptions of infinities. It all seems rather predicated on shaky premises.
Much of that stuff can be derived without any shaky (IMO) premises.  The basic idea is that two sets are considered to be of the same "size" if there exists a one-to-one correspondence between them.  That's how we compare finite sets, so why not extend the same idea to infinite sets?

And if you make an arbitrary one-to-one function from the set of natural numbers into the set of real numbers, one can always find a real number that is outside the range of that function.  So such a one-to-one correspondence does not exist.  And therefore the set of real numbers is bigger than the set of natural numbers.  Nothing shaky there AFAICS.
also, while I can understand why people initially think the idea of a "countably infinite" set is nonsensical or a useless abstract curiosity (if it's infinite, then obviously you will never actually count it), it does have practical value. 

e.g., if you're writing an algorithm that needs to find a particular element of a set (e.g., a particular integer or rational solution of some equation (which is known to have a solution, specifically an integer or rational one)), you can write said algorithm so that it checks "all" elements of the set (more precisely, never skips any), and you can rest assured it will find the solution in a finite amount of time. 

if the set you're working with is uncountably infinite, I believe that is not true.  it is not possible to individually check each and every of the reals; and if you try, there is no guarantee (and a very low chance) you will find one that satisfies a particular condition, in any finite amount of time (however long), even if it is known to exist.  ...  more generally, iiuc, it is impossible to execute some operation, "for each" of the reals, even if restricted to some finite interval.
  • Last Edit: September 09, 2016, 08:46:11 AM by el jefe

  • 52

  • 22

Re: infinities and the mind-dependence of maths (split from Newton etc.)
Reply #107
More specifically, suppose that P(x) is a predicate in the language of some suitably-chosen mathematical theory T. If there is a non-constructive proof of the statement "there exists A such that P(A)" but demonstrably no constructive proof, then we're forced to admit the existence of objects A such that P(A) while nevertheless holding that there are no examples of such objects (for an example would yield a constructive proof!).
Out of curiosity, does this (the highlighted situation) ever happen in practice?

All the time. For example, the intermediate value theorem is an existence result with no constructive proof (at least as it's stated in most analysis texts).

Quote
Quote from: Quizalufagus
This is an awkward position to be in as a platonist because there must be an objective matter of fact about whether such an A "exists" for a pretty naively common-sense notion of what it means to "exist." As far as I can see, the only way a platonist has around this difficulty is by claiming that the theory T doesn't actually adequately capture everything there is to know about the subject matter of T, and that there is indeed an objective matter-of-fact about that subject matter that we'll never know. That seems pretty mystical when we're talking about
More specifically, suppose that P(x) is a predicate in the language of some suitably-chosen mathematical theory T. If there is a non-constructive proof of the statement "there exists A such that P(A)" but demonstrably no constructive proof, then we're forced to admit the existence of objects A such that P(A) while nevertheless holding that there are no examples of such objects (for an example would yield a constructive proof!).
Out of curiosity, does this (the highlighted situation) ever happen in practice?
Quote from: Quizalufagus
This is an awkward position to be in as a platonist because there must be an objective matter of fact about whether such an A "exists" for a pretty naively common-sense notion of what it means to "exist." As far as I can see, the only way a platonist has around this difficulty is by claiming that the theory T doesn't actually adequately capture everything there is to know about the subject matter of T, and that there is indeed an objective matter-of-fact about that subject matter that we'll never know. That seems pretty mystical when we're talking about abstract objects.
Hmmm.  "Pretty mystical" or not, it doesn't clash with my instincts at all.  I don't see a problem here.

You mean the potential fix that I've suggested? It doesn't clash with my instincts either as long as we're talking about relatively down-to-earth objects like numbers. But it's hard for me to believe that there's an objective platonic realm of (for example) C*-algebras and that our axioms for them are just an incomplete attempt at capturing an elusive part of nature that's already there.

  • 7,305

  • 1092

Re: infinities and the mind-dependence of maths (split from Newton etc.)
Reply #108
I'll get back to this probably tomorrow  broD. It drifted some so I'll go back to the beginning and try to pick up where my argument really was. At this point, it's gotten into some sort of exposition of entire philosophies which probably aren't that worked out to be subject to this level of scrutiny. :)
Love is like a magic penny
 if you hold it tight you won't have any
if you give it away you'll have so many
they'll be rolling all over the floor

  • Brother Daniel
  • Global Moderator
  • predisposed to antagonism
  • 605

  • 160

Re: infinities and the mind-dependence of maths (split from Newton etc.)
Reply #109
More specifically, suppose that P(x) is a predicate in the language of some suitably-chosen mathematical theory T. If there is a non-constructive proof of the statement "there exists A such that P(A)" but demonstrably no constructive proof, then we're forced to admit the existence of objects A such that P(A) while nevertheless holding that there are no examples of such objects (for an example would yield a constructive proof!).
Out of curiosity, does this (the highlighted situation) ever happen in practice?
All the time. For example, the intermediate value theorem is an existence result with no constructive proof (at least as it's stated in most analysis texts).
Sure, I'm fairly familiar with the typical non-constructive proofs of the intermediate value theorem.  But I'm not at all familiar with any proof of the nonexistence of constructive proofs thereof.  Does a proof of the latter sort appear in most analysis texts?
Quote from: Quizalufagus
You mean the potential fix that I've suggested?
Yes.
Quote from: Quizalufagus
It doesn't clash with my instincts either as long as we're talking about relatively down-to-earth objects like numbers. But it's hard for me to believe that there's an objective platonic realm of (for example) C*-algebras and that our axioms for them are just an incomplete attempt at capturing an elusive part of nature that's already there.
Not sure I see a relevant difference here.

  • 52

  • 22

Re: infinities and the mind-dependence of maths (split from Newton etc.)
Reply #110
Sure, I'm fairly familiar with the typical non-constructive proofs of the intermediate value theorem.  But I'm not at all familiar with any proof of the nonexistence of constructive proofs thereof.  Does a proof of the latter sort appear in most analysis texts?

Nope, sadly most aspects of constructive analysis are quite arcane (and it's certainly not my area, either). I seem to recall reading something in grad school that states that the usual IVT actually implies the law of the excluded middle, however, although that's probably sensitive to particular choices about how to build the reals constructively. There's an edifying discussion here that at least shows the usual IVT is provably unprovable in constructive analysis.

  • Pingu
  • Needs a Life
  • 7,107

  • 1064

Re: infinities and the mind-dependence of maths (split from Newton etc.)
Reply #111
Hi Quiz!

  • 52

  • 22

Re: infinities and the mind-dependence of maths (split from Newton etc.)
Reply #112
Hi Lizzie.  :)

  • 7,305

  • 1092

Re: infinities and the mind-dependence of maths (split from Newton etc.)
Reply #113
Take a set of all the rational numbers. That is an operation which encapsulates an infinity. Then you can say that the set cannot be a member of itself so you get into the neverending recursive loop. But before you took that set, it didn't exist. So when you took that set, you created something which is iterative. If you want to do it again, then you need to include the previous set but only if you want to do it again. The iterative nature of the problem is the problem. The idea of infinity assumes a god's eye view which simply doesn't exist anywhere, not even in maths.
I can't begin to make head or tail of any of this.
Since this is where I went off the rails, Ima start here.
Quote

I don't know what you mean by "taking" a set.  If I tell someone to "take" a set, I'd just mean "consider" that set (because I'm going to be talking about it).  How is that an "operation"?
By considering a set, it is brought into being. Because a set is defined by the axioms of set theory and its particular parameters, a set is a definition of some number space. That definition tells another person how to choose the elements that will go in the set. That requires an operation each and every time. All you have is rules until you have something to execute them. Does a hank williams record have there's a tear in my beer on it or is that a product of the interaction of the record and the record player? The groove is the instructions that the needle obeys.

Quote
What does it mean to "encapsulate" an infinity?
In this case, I meant to define the members of an infinite set. "The rationals". for example.
Quote

What recursive loop are you talking about?
Russell's paradox, whether it can be avoided by different axioms or not, is accurately summarized by saying you cannot have a set that includes all sets because that set would need to be a member of itself. Using just the logic given in that statement, if you were to add that set, you would make another set as the product. To actively try to construct a set that includes all sets (despite its futility), you would have to begin a process of continuously making new sets including the previous set. The obvious problem, that there is no obvious way to stop this pattern once you've started it, is why we call it a paradox. To attempt to carry out the instruction, we enter what we would recognize as an infinite set of iterations because the process has no rule which would make it stop at some future point.

Does that part make sense?.

Quote
I don't see how the existence of a set can have anything to do with what I happen to be doing.  When I "take" a set (in my sense of the word), I'm just considering something that's there anyway, I'm not creating anything.

How does infinity assume a god's eye view?  I'm not a god, and I can think about infinite sets without any problem.
I will get to the god's eye view once I figure out if my first part is just a dumb misunderstanding or is a legitimate issue.
Love is like a magic penny
 if you hold it tight you won't have any
if you give it away you'll have so many
they'll be rolling all over the floor

  • 7,305

  • 1092

Re: infinities and the mind-dependence of maths (split from Newton etc.)
Reply #114
Interesting:
http://nautil.us/blog/how-a-hypothesis-can-be-neither-true-nor-false
Quote
But it was not just rejection by Kronecker that pushed Cantor to depression; it was his inability to prove a particular mathematical conjecture he formulated in 1878, and was convinced was true, called the Continuum Hypothesis. But if he blamed himself, he did so needlessly. The debate over the conjecture is profoundly uncertain: in 1940 Kurt Gödel proved that the Continuum Hypothesis cannot be disproven (technically speaking, that the negation of the Hypothesis cannot be proven), and in 1963 Paul Cohen proved that it cannot be proven. Poor Cantor had chosen quite the mast to lash himself to.

    In his first known bout of depression, Cantor wrote 52 letters to the Swedish mathematician Gösta Mittag-Leffler, each of which mentioned Kronecker.

How is it possible, though, for something to be provably neither provable nor disprovable? An exact answer would take many pages of definitions, lemmas, and proofs. But we can get a feeling for what this peculiar truth condition involves rather more quickly.

Cantor's Continuum Hypothesis is a statement regarding sizes of infinity. To see how infinity can have more than one size, let's first ask ourselves how the sizes of ordinary numbers are compared. Consider a collection of goats in a small forest. If there are six goats and six trees, and each goat is tethered to a different tree, then each goat and tree are uniquely paired. This pairing is called a "correspondence" between the goats and the trees. If, however, there are six goats and eight trees, we will not be able to set up such a correspondence: no matter how hard we try, there will be two trees that are goat-free.

Correspondences can be used to compare the sizes of much larger collections than six goats--including infinite collections. The rule is that, if a correspondence exists between two collections, then they have the same size. If not, then one must be bigger. For example, the collection of all natural numbers {1,2,3,4,...} contains the collection of all multiples of five {5,10,15,20,...}. At first glance, this seems to indicate that the collection of natural numbers is larger than the collection of multiples of five. But in fact they are equal in size: every natural number can be paired uniquely with a multiple of five such that no number in either collection remains unpaired. One such correspondence would involve the number 1 pairing with 5, 2 with 10, and so on.

If we repeat this exercise to compare "real" numbers (these include whole numbers, fractions, decimals, and irrational numbers) with natural numbers, we find that the collection of real numbers is larger. In other words, it can be proven that a correspondence cannot exist between the two collections.

The Continuum Hypothesis states that there is no infinite collection of real numbers larger than the collection of natural numbers, but smaller than the collection of all real numbers. Cantor was convinced, but could never quite prove it.

To see why, let's begin by considering what a math proof consists of. Mathematical results are proven using axioms and logic. Axioms are statements about primitive mathematical concepts that are so intuitively evident that one does not question their validity. An example of an axiom is that, given any natural number (which is a primitive concept), there exists a larger natural number. This is self-evident, and not in serious doubt. Logic is then used to derive sophisticated results from axioms. Eventually, we are able to construct models, which are mathematical structures that satisfy a collection of axioms.

Crucially, any statement proven from axioms, through the use of logic, will be true when interpreted in any model that makes those axioms true.

    If there are six goats and eight trees, we will not be able to set up such a correspondence: no matter how hard we try, there will be two trees that are goat-free.

It is a remarkable fact that all of mathematics can be derived using axioms related to the primitive concept of a collection (usually called a "set" in mathematics). The branch of mathematics that does this work is known as set theory. One can prove mathematical statements by first appropriately interpreting the statement in the language of sets (which can always be done), and then applying logic to the axioms of sets. Some set axioms include that we can gather together particular elements of one set to make a new set; and that there exists an infinite set.

Kurt Gödel described a model that satisfies the axioms of set theory, which does not allow for an infinite set to exist whose size is between the natural numbers and the real numbers. This prevented the Continuum Hypothesis from being disproven. Remarkably, some years later, Paul Cohen succeeded in finding another model of set theory that also satisfies set theory axioms, that doesallow for such a set to exist. This prevented the Continuum Hypothesis from being proven.

Put another way: for there to be a proof of the Continuum Hypothesis, it would have to be true in all models of set theory, which it isn't. Similarly, for the Hypothesis to be disproven, it would have to remain invalid in all models of set theory, which it also isn't.

It remains possible that new, as yet unknown, axioms will show the Hypothesis to be true or false. For example, an axiom offering a new way to form sets from existing ones might give us the ability to create hitherto unknown sets that disprove the Hypothesis. There are many such axioms, generally known as "large cardinal axioms." These axioms form an active branch of research in modern set theory, but no hard conclusions have been reached.

The uncertainty surrounding the Continuum Hypothesis is unique and important because it is nested deep within the structure of mathematics itself. This raises profound issues concerning the philosophy of science and the axiomatic method. Mathematics has been shown to be "unreasonably effective" in describing the universe. So it is natural to wonder whether the uncertainties inherent to mathematics translate into inherent uncertainties about the way the universe functions. Is there a fundamental capriciousness to the basic laws of the universe? Is it possible that there are different universes where mathematical facts are rendered differently? Until the Continuum Hypothesis is resolved, one might be tempted to conclude that there are.


Ayalur Krishnan is an Assistant Professor of Mathematics at Kingsborough CC, CUNY.
Love is like a magic penny
 if you hold it tight you won't have any
if you give it away you'll have so many
they'll be rolling all over the floor

  • Brother Daniel
  • Global Moderator
  • predisposed to antagonism
  • 605

  • 160

Re: infinities and the mind-dependence of maths (split from Newton etc.)
Reply #115
By considering a set, it is brought into being. Because a set is defined by the axioms of set theory and its particular parameters, a set is a definition of some number space. That definition tells another person how to choose the elements that will go in the set. That requires an operation each and every time.
"Each and every time" is too vague.  Each and every time what?
Quote from: Testy
All you have is rules until you have something to execute them.
"All you have ... until":  Again you're implicitly conceding the existence of something before the "until".  In this case "rules".  OK then, let's define a set as the theoretical product of following those "rules".  Then the existence of the set is on the same footing as the existence of the rules themselves, which (in your own account) is execution-independent.
Quote from: Testy
Does a hank williams record have there's a tear in my beer on it or is that a product of the interaction of the record and the record player?
Both/and.
Quote from: Testy
Russell's paradox, whether it can be avoided by different axioms or not, is accurately summarized by saying you cannot have a set that includes all sets because that set would need to be a member of itself. Using just the logic given in that statement, if you were to add that set, you would make another set as the product. To actively try to construct a set that includes all sets (despite its futility), you would have to begin a process of continuously making new sets including the previous set. The obvious problem, that there is no obvious way to stop this pattern once you've started it, is why we call it a paradox. To attempt to carry out the instruction, we enter what we would recognize as an infinite set of iterations because the process has no rule which would make it stop at some future point.

Does that part make sense?.
Not at all, AFAICS.

First of all, you've misidentified Russell's Paradox (which is surprising, considering that you quoted wiki about it earlier, but seem to have ignored what you quoted).  There's nothing inherently paradoxical about having a set be an element of itself, in general.  The paradox arises when you try to partition the set of all sets into the two sets E1 = {x | x is a set and xx} and E2 = {x | x is a set and xx}, and then try to figure out whether E2 itself belongs to E1 or to E2.  You get a contradiction either way.  That is the paradox.

Second, your description of the alleged difficulty of having a set of all sets is analogous to rejecting an equation as insoluble if the desired variable happens to fall on both sides of the equation.  There's nothing wrong with saying "let x be the negative real number having the property that x = 1/(x - 1)".  Iterative methods can be useful, but self-referentiality does not inherently create iterative-ness.

  • 7,305

  • 1092

Re: infinities and the mind-dependence of maths (split from Newton etc.)
Reply #116
1.Every time you want to define a set.

2. generating the rules is an operation too.

3. "Both/and" We have a very different way of conceptualizing this. I disagree with your view. That is the crux of the issue.

4. I haven't misidentified the paradox. I reframed it. If that reframing bothers you (and I get why it would) then you can go back to my earlier post where I used it in the normal way and explained the temporal quality that creates it. It is not the same as two solutions though. Set E2 is not available when E1 is constructed. The paradox results from moving outside of the iterative process.

Love is like a magic penny
 if you hold it tight you won't have any
if you give it away you'll have so many
they'll be rolling all over the floor

  • Brother Daniel
  • Global Moderator
  • predisposed to antagonism
  • 605

  • 160

Re: infinities and the mind-dependence of maths (split from Newton etc.)
Reply #117
1.Every time you want to define a set.
Suppose I write a neat proof of some mathematical proposition. Suppose my proof starts with "Let X = ..." and defines a set, giving it the name X.  Have I done this "operation" just once, in the act of writing?  Do you do this "operation" yourself every time you read my proof?  Or just every time you read it carefully?  Or suppose that the proof is lengthy, and that you forget what X is part way through, and you have to glance back at the first sentence to see what X is again.  Are you repeating the "operation" and creating another new set in that case?  On the other hand, suppose the proof is brief, and you don't forget it at all:  Are you then not repeating the "operation" if you reread the proof immediately after reading it the first time?  I just can't see a coherent account of when exactly the "operation" takes place in your view.
Quote from: Testy
2. generating the rules is an operation too.
And you're generating those rules by executing some meta-rules.  And generating those meta-rules by executing some meta-meta-rules.  And so on ad infinitum.  So every time you generate a set, you have an infinite regress of "operations", each of which, being a mental process, takes at least some minimum finite amount of time.  So each of us must be infinitely old.
Quote from: Testy
3. "Both/and" We have a very different way of conceptualizing this. I disagree with your view. That is the crux of the issue.
Yup.
Quote from: Testy
4. I haven't misidentified the paradox. I reframed it.
Rubbish.  Your latest alleged summary of Russell's Paradox has nearly nothing to do with what Russell's Paradox actually is.  You've (at best) identified a completely different issue - though I'm not convinced that the issue you've identified is really an issue at all, much less a paradox.
Quote from: Testy
If that reframing bothers you (and I get why it would) then you can go back to my earlier post where I used it in the normal way and explained the temporal quality that creates it.
That earlier post was completely garbled and incoherent, though.  Well, the quotation from wiki made perfect sense, but I can't see any way in which you "used" it or "explained" anything about it.

You haven't given a coherent explanation of why there has to be an iterative process involved.  You haven't given a coherent argument for why, if the set of all sets must be "created" by invoking it, the creation can't work in one swell foop.

  • 7,305

  • 1092

Re: infinities and the mind-dependence of maths (split from Newton etc.)
Reply #118
rubbish. You just haven't accepted my premises.
Love is like a magic penny
 if you hold it tight you won't have any
if you give it away you'll have so many
they'll be rolling all over the floor

  • 7,305

  • 1092

Re: infinities and the mind-dependence of maths (split from Newton etc.)
Reply #119
rubbish. You just haven't accepted my premises.
Actually, this is what I find odd about this discussion. You are saying that you don't understand when what you mean is either that you disagree with my premises or that you are really blinded by your own to the extent that you require them to be accepted before you will engage your mind. The latter has a mildly Morton's Demonesque undercurrent (not an insult, you could call it lensing if you wanted and anyway I don't know if it's what's going on here or not).

I don't have enough invested in my premises to really fight about them. They are just premises about a classically non-instrumental topic, navel gazing so to speak. But I can easily understand your premises and articulate why I disagree with them. Mine are simply that mind is where meaning is made, where perception is turned to information, and it's made in discrete events. If you begin with those premises, the rest just follows.
Love is like a magic penny
 if you hold it tight you won't have any
if you give it away you'll have so many
they'll be rolling all over the floor

  • 161

  • 41

Re: infinities and the mind-dependence of maths (split from Newton etc.)
Reply #120
Testy: you have misstated the paradox. There is nothing in ZFC without the axiom of foundation that contradicts the existence of sets that contain themselves. You can have sets like x = \{x\}.

eta: Where's the latex here?

  • el jefe
  • Needs a Life
  • asleep till 2020 or 2024
  • 2,314

  • 479

Re: infinities and the mind-dependence of maths (split from Newton etc.)
Reply #121
some people are allergic

  • Brother Daniel
  • Global Moderator
  • predisposed to antagonism
  • 605

  • 160

Re: infinities and the mind-dependence of maths (split from Newton etc.)
Reply #122
Actually, this is what I find odd about this discussion. You are saying that you don't understand when what you mean is either that you disagree with my premises or that you are really blinded by your own to the extent that you require them to be accepted before you will engage your mind. The latter has a mildly Morton's Demonesque undercurrent (not an insult, you could call it lensing if you wanted and anyway I don't know if it's what's going on here or not).
Looks insulting to me.  But that's ok, I've probably insulted you a few times itt, and will probably do so a few more times (even in this very post).  I still like you anyway.  :)

Anyway, no.  I say that much of what you've said makes no sense at all - not that it's "wrong", because you have to start making sense before you can even be wrong.  And to the extent that I can pull some sense out of your words (often remaining unsure that you're saying what I think you're saying), your (apparent) assertions remain unsupported.  It has nothing to do with disagreement with your premises; I'm perfectly happy to take anyone's premises for the sake of argument and see where they go.  There's no refusal to engage my mind.

It may be that I'm simply not smart enough for this discussion.  I suspect that that often happens, that I get into a disagreement with someone who has insights that I simply can't grasp.  But in this case, I'm given a pretty strong clue that something else is going on, by your pigheaded insistence on remaining manifestly wrong about what Russell's Paradox is about.  The more likely problem, it seems, is that your thoughts are as incoherent as your words.  (There, I think I just insulted you.)
Quote from: Testy
But I can easily understand your premises and articulate why I disagree with them.
Are you going to start doing that, then?
Quote from: Testy
Mine are simply that mind is where meaning is made, where perception is turned to information, and it's made in discrete events. If you begin with those premises, the rest just follows.
I don't have a problem with those premises as stated.  It's not clear to me that they clash with anything I've said.  I don't agree that "the rest just follows".

Take, for example, your notion that self-referentiality in a set definition necessarily indicates some sort of iterative process.  AFAICS, you haven't even begun to try to show how that follows from your premises.