Fractals use fractional dimensions... @Metaphysician Undercover can't go there.

Meta's queer usage. — Banno

As you noted, we're all a little odd sometimes, just as we're all a little, actually a lot, ignorant most of the time. Ignorance yields to knowledge and information, learns in other words, else it becomes just stupidity - and some of us even arrive there from time to time. But beyond these informal divisions it gets dark quickly. Beyond the border of stupid on the other side is evil. Why? For what reason? To what end? That's often not immediately evident. And it may be the world we find ourselves in at the moment, but I am increasingly inclined to call out evil masquerading as stupidity (and masquerading as anything else), and to give that evil little or no space, or air to breathe.

@Metaphysician Undercover has been engaged and challenged in this and many other threads by many people who have taken him on in good faith on a series of outrageous, ignorant, indefensible claims that he has made. At some point charity facilitates and enables, and at that point being a nice guy is an error.

I have no desire to waste time on MU's nonsense to date. I commend to him though the advantages and benefits for all if he's got his arguments tested and in order going forward, else he find the going that he makes difficult for everyone else, equally as difficult for himself.

He can start here: according to MU real numbers, with the exception of integers, are not numbers, and:

A true square does not admit to a diagonal, the two sides are incommensurable, making the square an irrational figure, just like the circle. There is no such thing as the diagonal of a square, because there is no such thing as a square, just like there's no such thing as a circle. — Metaphysician Undercover

In as much as he, or anyone, can believe what they like, still, they ought not present their beliefs as categorically true. I invite him to here rigorously prove his claims, or recant and retire in silence.

I wonder how efforts to attract a professional philosopher to this forum are progressing? :roll:

I mentioned before - I don't know if you noticed it - that this thread is not about mathematics so much as about the psychology of crackpots, of which Meta is certainly one.

i've been engaging with Meta for years - his posts are like a broken tooth that one keeps probing with one's tongue.Perhaps there is something to be gained here, not by treating Meta's posts seriously, but by looking at how he avoids confronting the truth.

For example he provides a restricted definition that suits his purposes, and when challenged he demands 'I invited anyone to provide a better definition of "number"'; a "have you stoped beating your wife" response.

He engages in pedantry for rhetorical purposes, as when he avoided @Pfhorrest's point concerning cones, and when he dithers on fractions being quantities, and pretending to @fishfry that his argument pivots on the distinction between equivalent and identical.

When met with a refutation he will deny it and simply repeat the refuted argument.

Then there is the paranoia, as in his reply to you that "Modern mathematics contains a lot of sophistry, of which some is used for deception".

Again, there is the outstanding point that he fails to directly address the two arguments presented in the OP. I think this is in order to avoid rigour.

And there is an extraordinary lack of self-awareness, of just how far he has deviated from orthodoxy, and how much of mathematics he must reject. Of course, for him this might be seen as proof of his intellectual courage, his rugged individuality.

Elsewhere his lack of comprehension has derailed whole threads - repeatedly, in the case of discussions of Wittgenstein on rules and on private language.

But is he evil? I don't think so, although you rightly question his motivation. I would suggest that it's not malevolence; but that he has found that adopting this approach brings attention. He is, probably only semi-consciously, espousing extremes in order to get a reaction.

And it works.

:wink:

Not so well. Can't think why.

They've proved it. — Michael

They think they've proved it. Staring from a false premise does not make a sound proof. But if one doesn't recognize the falsity of the premise...

A, therefore B, where A is .999... = 1, and B is some rambling about equivalence. But here we don't merely have equivalence, we have equality. Because we have equality, they do represent the same value. I've never heard of someone so far gone as to commit an amphiboly by changing the word. But in this post, and here?: — InPitzotl

"1" represents a value which is a quantity. No one has demonstrated how 1/9 represents any particular quantity, because as I've explained, it does not represent a quantity. So how do you claim they are equal or equivalent?

That definition was already discussed in the thread. And that definition is used in the pdf provided by the op in section 1. By that definition, .999... = 1 exactly. — InPitzotl

I see no such definition. Perhaps you can produce that definition so that we can determine whether 1/9, and .111... represent numbers. According to the definition of "number" which I provided they do not represent numbers. Where's your definition of "number"?

n other words, 65 pennies, a dime and a quarter is not worth a dollar because pennies are 1/100th of a dollar and that's not a particular quantity of money. I mean, sure, some pennies are smaller than other pennies slightly; but some dollar coins are also smaller than other dollar coins. But apparently the pennies being smaller implies that pennies aren't a particular value, whereas the dollar coins being smaller does not indicate such a thing. Such is the tomfoolery I've heard from you so far. That's a garbage argument that can be ignored just on its merits. — InPitzotl

Sorry, I have no idea of what you're talking about again. I wish you could make a greater effort to make clear what you want to say.

Sure they do. 1/2 represents one half. As you said, one of anything represents a particular quantity. The quantity that half represents is very clear... that is the multiplicative inverse of two. It takes two halves to make the quantity one. — InPitzotl

I don't see how you can say that one half represents any particular quantity, when one half of two is different from one half of four which is different from one half of eight, etc.. The quantity represented by "one half" is clearly, and completely dependent on the context. So how could "one half" on its own, represent any particular quantity.

So, you claim that there is a quantity called one half. That's nonsense. There is no such thing as one half, unless it is a half of something, and that "something", which is required of necessity for the existence of the half, sets the value for the quantity which the half is a half of. If you really think that there is such a thing as a half which is not a half of something, and is an actual quantity all on its own, then show it to me, give me an example. Show me a half which is not a half of something. It's impossible, because "half" is by definition half of something. I'm sure you must really understand this though, that there is no such thing as a half which is not a half of something, and that one half cannot have any quantitative value whatsoever unless it is stated what it is a half of. Are you just playing dumb?

And no, it does not take two halves to make one, that's a falsity. Two halves are made by dividing one. Two equal things together, are two, not one. And one is a unity which is not necessarily made from two halves. I'm shocked that you are unfamiliar with these fundamental principles.

The way mathematics speakers use the term "particular quantity", 1/9 is indeed one of those things. — InPitzotl

Ha ha, that's ridiculous. 1/9 can be any quantity you want, depending on the size of the whole which is being divided nine ways. 1/9 can be one if the whole is nine, it can be two if the whole is eighteen, it can be three if the whole is 27, it can be four if the whole is thirty six, and so on and so forth. The idea that 1/9 itself, is a particular quantity is utter lunacy. And if mathematics speakers really use the term in this way, then I'd have to say that they really do not know what they're talking about. Don't you agree? How can something (1/9), which can be absolutely any quantity whatsoever, be said to be a particular quantity? And how can you not see the ridiculousness of the claim that it is a particular quantity?

Therefore, your real burden is to show what's wrong with the language of math; you can't just say, "I don't 'believe' 1/9 is a particular quantity"... you have to say, "saying '1/9' is a particular quantity leads to the following problem" and say what that problem — InPitzotl

I've already demonstrated this, numerous times already now. It's utter nonsense to insist that 1/9 is any particular quantity, when it's very clear that it can be any quantity whatsoever. And please don't suggest as Banno did, that 1/9 really means one divided in nine equal parts, because that's something impossible.

Well, Metaphysician Undercover hasn't addressed the two proofs from the OP. All he has done is to assert that 1/9, and other fractions, are not numbers. His argument is an appeal to the authority of the OED. — Banno

I'm still waiting for a better definition of "number". "Family Resemblance" doesn't suffice in logic.

ncidentally, and to my great amusement, the OED definition of fraction is "...numerical quantity that is not a whole number...", contradicting Meta's assertion that fractions are nether numbers nor quantities. — Banno

Actually, this doesn't contradict my claim at all. It just shows that the definitions of mathematicians contradict themselves. That's the problem with mathematics, which I will not cease to demonstrate, it's loaded with contradictions.

Perhaps there is something to be gained here, not by treating Meta's posts seriously, but by looking at how he avoids confronting the truth. — Banno

Let's just say that there is something to be gained from looking at how people, in general, avoid confronting the truth, and not single out any individuals here. Can we leave it at that? Or are you so absolutely certain that what you claim is the truth, and what I claim is false, that you would single out me as the one who is so certain?

For example he provides a restricted definition that suits his purposes, and when challenged he demands 'I invited anyone to provide a better definition of "number"'; a "have you stoped beating your wife" response. — Banno

The op clearly deals with "a matter of representing numbers". If it is the case that some of the symbols used in the op do not actually represent numbers, then we have a false representation. Therefore we require a definition of "number" to determine whether or not there is such a false representation. I provided a definition of "number". According to this definition of "number" we have a false representation in the op.

You are not willing to accept the truth of this, so you reject my definition. Now we have no definition of "number", and no way to resolve the question of whether the op gives us a false representation. Therefore I implore you to provide a better definition of "number", so that we can truly see whether there is a false representation or not, resolve this issue to everyone's satisfaction, and get on to something less trivial.

Again, there is the outstanding point that he fails to directly address the two arguments presented in the OP. I think this is in order to avoid rigour. — Banno

There is no point to addressing the arguments themselves until we determine whether or not there is a false premise. Is this a matter of representing numbers or not. I think it's very clearly not such a matter. You disagree. Where's your argument? What is your criteria for "a matter of representing numbers", which makes you so strongly believe that this is actually a matter of representing numbers?

I see what is expressed in the op as a matter of dividing magnitudes. And, it has been demonstrated numerous times, over and over again throughout history, that some theoretical divisions cannot be represented in number. Claiming to have a numerical representation of what cannot be represented in number, is clearly a false claim. Don't you agree?

It just shows that the definitions of mathematicians contradict themselves. — Metaphysician Undercover

:lol:

@Metaphysician Undercover, out of curiosity, what do you make of i? I assume that for you imaginary numbers are not numbers.

Redacted. Move on to Tim's question.

He can start here: according to MU real numbers, with the exception of integers, are not numbers, and:
A true square does not admit to a diagonal, the two sides are incommensurable, making the square an irrational figure, just like the circle. There is no such thing as the diagonal of a square, because there is no such thing as a square, just like there's no such thing as a circle.
— Metaphysician Undercover

In as much as he, or anyone, can believe what they like, still, they ought not present their beliefs as categorically true. I invite him to here rigorously prove his claims, or recant and retire in silence. — tim wood

Hey @Metaphysician Undercover! You didn't see this? Put up or shut up!

As a matter of representing numbers, wouldn't most be fine with 9/9 = 9 × (1/9) = 9 × (0.111...) ? — jorndoe

@Metaphysician Undercover, perhaps in answer to Tim's question you might set out where the flaw is in this calculation - regardless of wether the items involved are numbers or not, where in your view does this go wrong?

There is no such thing as one. Everything is infinite.

No one has demonstrated how 1/9 represents any particular quantity, — Metaphysician Undercover

Wrong. I have. Also, Banno has:

Incidentally, and to my great amusement, the OED definition of fraction is "...numerical quantity that is not a whole number...", contradicting Meta's assertion that fractions are nether numbers nor quantities. — Banno

...using your own dictionary. And your dictionary has, as demonstrated by Banno. Your problem is that you don't understand the language; that's confounded by the fact that you think you do.

because as I've explainedasserted, it does not represent a quantity.

FTFY. A ninth is the specific particular quantity corresponding to dividing one into nine equal units. That's why your same dictionary that you quoted the definition of a number in says that a fraction is a number.

So how do you claim they are equalor equivalent? — Metaphysician Undercover

By proof, such as the one given in the OP.

I see no such definition. — Metaphysician Undercover

We're using your OED definition.

According to the definition of "number" which I provided they do not represent numbers. Where's your definition of "number"? — Metaphysician Undercover

Wrong. According to that definition, they are numbers. You just don't understand that definition... see above.

Sorry, I have no idea of what you're talking about again. I wish you could make a greater effort to make clear what you want to say. — Metaphysician Undercover

That's quite interesting. What I was saying here is a direct analog of your points about fractions and pie applied to money according to my best assessment of what gibberish you're trying to push. So if you yourself don't understand this, maybe you should heed the advice you're trying to give me.

Ha ha, that's ridiculous. — Metaphysician Undercover

It's only ridiculous to you, because you don't speak the language.

1/9 can be any quantity you want, depending on the size of the whole which is being divided nine ways. — Metaphysician Undercover

We've already addressed this... you're saying nothing about fractions that doesn't also apply to counting numbers. To have a point you must special plead it.

1/9 can be one if the whole is nine, it can be two if the whole is eighteen, it can be three if the whole is 27, it can be four if the whole is thirty six, and so on and so forth. — Metaphysician Undercover

What are you talking about? A whole pie is one pie, not nine pies, eighteen pies, or twenty seven pies. You mean groups. Taking a particular quantity of equal sized groups is just multiplication. If I were at a farmer's market and they had a carton of a dozen eggs, I might could barter getting one half of a dozen. He'll give me six eggs. Or maybe I need more... maybe I need two dozens. He'll give me 24 eggs. Even your precious one dozen is twelve eggs. You're choking on multiplication.

And if mathematics speakers really use the term in this way, then I'd have to say that they really do not know what they're talking about. Don't you agree? — Metaphysician Undercover

Nope. I would say you had some severe misunderstanding of math.

How can something (1/9), which can be absolutely any quantity whatsoever, be said to be a particular quantity? — Metaphysician Undercover

1/9 is only a ninth of 1. But you can take 1 of anything, including groups.

And how can you not see the ridiculousness of the claim that it is a particular quantity? — Metaphysician Undercover

Because I understand how it makes sense, because I understand it.

May the farce be with you.

I think the whole .9999... = 1 thing is much more interesting to psychologists than to mathematicians.

I take my definition of "number" from OED: "an arithmetical value representing a particular quantity and used in counting and making calculations". — Metaphysician Undercover

It's a slow day, so I dug out both the Concise and the Shorter OED.

Neither contains Meta's definition.

How is that to be explain'd?

The first definition, given in both with slightly differing wording, is

The sum or aggregate of any collection of individual things or persons...

Followed by

A sum or total of abstract units...

and then...

the particular mark or symbol, having an arithmetical value...

and then on to other related uses.

It's a slow day, so i dug out both the Concise and the Shorter OED. — Banno

I found it on some random russian vocabulary site: (https://slovar-vocab.com/english/fundamental-vocab/number-6810737.html) (ETA: Unhiding this link). An expanded version can be found here.

That ain't no OED.

SO what do we decide - did Meta lie, or was he misled?

That ain't no OED. — Banno

The former has no attribution. The latter says "powered by Oxford" (OUP). I personally was granting that maybe he had one of those ancient analog thingies made of trees; can't quite trace it further than that (though TBH I didn't try too hard past those two).

I personally was granting that maybe he had one of those ancient analog thingies made of trees. — InPitzotl

That's what I was using, so that's no excuse.

Lexico claims "All definitions and translations are written by Oxford lexicographers". So we might give him the benefit of our doubt and conclude that he was misled rather than malicious.

I mentioned before - I don't know if you noticed it - that this thread is not about mathematics so much as about the psychology of crackpots, of which Meta is certainly one. — Banno

I only really know enough about maths to enjoy the explanations given here, not to provide any of my own, but I do know about the psychology of crackpots (we prefer to use the term 'nutjobs' nowadays, in these more enlightened times, unless the patient is allergic to nuts, in which case 'fruitcake' is fine).

What goes on here, I think, is that a person develops a fear of that which they do not understand. I think it's born of the extent to which we are no longer in control of our livelihoods (but that's another story entirely). The point is that philosophy-talk - the grammatical form of the arguments that philosophy uses - acts like heroin, in that it supplies a way of distancing oneself from that uncertainty and complexity. Rather than words being used, as they really are, to fumble about in the dark trying to get other people to act in ways we'd like, they turn into containers to bind stuff to, to pin it down and appear to stop it from being so ineffable.

Dictionary definitions are the drug dealers here. Supplying a few thousand such strong-boxes in which we can lock uncertain aspects of the world we experience. Numbers are no longer a complicated product of our minds, slightly intangible in places, occasionally contradictory if taken out of context, with some odd consequences we can quite get our heads around (like i). They now become what the dictionary says they are (usually ignoring the second and third definitions) - tamed and chained.

It's a weird flipping of our rational capacity from being that which tries to make sense of an already existent world to that which creates that world according to its rules, and thereby regains control over it. Instead of trying to work out some way of modelling what we experience, we simply claim to only experience that which we have modelled. A good proportion of the arguments here could be summed up as "It seems that way to me, therefore it must be that way". Of course, one acting this way needs a filler, something to explain all that which is beyond their current imagination - hence almost everyone using this drug is also religious - God of the Gaps.

Anyway, thanks to the mathematicians here who do patiently point out the flaws in these 'solipsistic' arguments, they do make interesting reading for an interested non-mathematician.

Cheers.

I suggested elsewhere that the motive might have been a search for recognition; perhaps this is transference on my part.

That link between the avoidance of ambiguity, political conservatism and religion is cogent.

I suggested elsewhere that the motive might have been a search for recognition — Banno

Yeah, I think that's true. I see it as part of the whole uncertainty issue. I mentioned this briefly on the Lazerowitz thread. Socially, the rise of science as powerful force, I think, results in a backlash rise in alternative 'expert' fields where they are immune to being demonstrably wrong. I think the same thing happens on a smaller scale here, and again is facilitated by the grammatical structure of philosophy-talk. Faced with a field in which it appears one can be demonstrably shown to be wrong, recognition is harder to come by and more fragile when attained. A simpler tactic is to set up an alternate set of rules and , regardless of their utility, raise oneself as an expert in those. One cannot be demonstrably wrong, one is instantly the world's foremost expert and one did not even have to leave one's armchair. Of, corse many see this as a hollow victory because the very public rejection of such rules is sufficient to pour cold water on any feelings of grandeur.

Many people are immune, or resistant, to updating their beliefs in the face of rejection by their peers. This is usually a good thing because it's how we get innovation and resistance to oppression etc. But again here philosophy-talk lets us down, it gives the impression that we can do this with language too - that we can reject the 'oppressive, conservative' use of terms to refer to A and insist they refer to B - ignoring the fact that language is a social endeavour, agreement is the substance of it, not a side-effect.

So yeah, I think you're right, people do maintain these very private structures as a kind of 'cheat mode' for the progression to recognition, and I think the means by which they do it is to mistake language for the kind of thing where innovation is bold and entrepreneurial and so become immune to the rest of their language community responding as if they were mad.

A further point is resistance to disproof. It's curious to see how Meta, who is articulate and intelligent, is able to resist recognising that his ideas have been falsified.

The argument, in its cleanest, most direct form, is given in the OP:

9/9 = 9 × (1/9) = 9 × (0.111...) — jorndoe

Meta did not directly address this, or any other such proof. Instead he went to an irrelevance, his claim that 1/9 is not a number.

Even if 1/9 were not considered a number, the proof would stand.

I suppose that for him it's not the mathematical process that is at fault so much as the picture of how mathematics works - for Meta, mathematicians have failed to understand their topic, and so their proofs are irrelevant. His task becomes not the rejection of the proof, but the correction of the mathematician's picture.

Hence, the mathematicians who engaged with him could never succeed in showing him how he went wrong; for Meta their very arguments are based on false premises. They may have thought that they were in a discourse about mathematics, but they were in a discourse about Meta's certainty.

They may have thought that they were in a discourse about mathematics, but they were in a discourse about Meta's certainty. — Banno

Yes, that's exactly it. Proving a theorem by the rules of mathematics is seen as irrelevant because the game is not to accept the rules and try to understand where they lead, the game is to take one's current understanding and construct rules which make it right.

It's obviously a triggered response at some point in their life. Children cannot develop at all if they were born with such an attitude, no understanding of anything would come about. It must therefore be something these people decide at some point in their life - "That's it, no more understanding, no more modelling, from now on the world has to change to fit what I already understand!".

What I can't decide is which came first. Whether the need to respond this way promotes that kind of philosophy, or whether that kind of philosophy entices people into responding that way. If there was no form of discourse in which one could appear to argue about what a number "really is", would they invent one to meet the need, or would they be out of alternatives and have to fumble by with only half understanding like the rest of us?

They think they've proved it. Staring from a false premise does not make a sound proof. But if one doesn't recognize the falsity of the premise... — Metaphysician Undercover

Which premise is false?

And don't you think that the professional mathematicians know more about maths than you do? I don't know what your area of expertise is but I'm pretty certain that it's something else.

What I can't decide is which came first. Whether the need to respond this way promotes that kind of philosophy, or whether that kind of philosophy entices people into responding that way. If there was no form of discourse in which one could appear to argue about what a number "really is", would they invent one to meet the need, or would they be out of alternatives and have to fumble by with only half understanding like the rest of us? — Isaac

Perhaps philosophy allows someone like Meta to hide what he is doing. If this were a thread about Wittgenstein, only those who had read Wittgenstein and the associated literature would recognise the misrepresentation in which Meat engages - as indeed did happen in the threads involving Sam (of blessed memory) and others.

It's the topic - mathematics is clear cut, and so it's harder to hide.

It's the topic - mathematics is clear cut, and so it's harder to hide. — Banno

When it's so close to the usual definitions, yeah.

Now you've struck the heart of the problem. Some quantities cannot be divided in certain ways. It is impossible. Three cannot be divided by nine, it is impossible. Nevertheless, mathemagicians are an odd sort, very crafty, wily like the fox, devising new illusions all the time. They like to demonstrate that they can do the impossible. Some people even believe that they actually do what is impossible. That is a problem. — Metaphysician Undercover

You didn't see this? Put up or shut up! — tim wood

I've already given my proof, based in my definition of "number". It's in my posts directed at Banno. Don't listen to Banno here because Banno's form of discussion is to pay no attention to what the other person says.

We can talk about making a theoretical division which is impossible to do, such as dividing one in nine equal parts, which is impossible to do. In reality it is contradictory to divide one into any parts, because then you are saying that it is not one, but however many parts you are dividing it into. Instead of recognizing that division of some quantities is impossible, some mathemagicians have proposed a new system of "numbers", which allows that impossible divisions can be represented as numbers.

The problem is that now they have so-called "numbers" which are outside the criteria of the definition of "number" (as presented by me), yet the mathemagicians provide no clear new definition of "number" which allows that these representations such as .111..., are actually representative of numbers. They give them a name "real numbers", so that they can refer to them, but the concept of "number" is just left vague, undefined, and full of inconsistencies.

Since I have provided a very clear definition of "number", and according to this definition many representations which are classed as real numbers are not actually numbers, so we can conclude that these real numbers are not actually numbers, based on that definition. If you want to demonstrate that my logic is unsound, I suggest you show me that my definition is false, by producing the true definition of "number", the one which allows that all real numbers are actually numbers. Otherwise we can look for a better name for things like 1/9, one which better represents what they are, such as "relations between quantities", or something like that .

perhaps in answer to Tim's question you might set out where the flaw is in this calculation - regardless of wether the items involved are numbers or not, where in your view does this go wrong? — Banno

I've already laid that out for you:

Now you've struck the heart of the problem. Some quantities cannot be divided in certain ways. It is impossible. Three cannot be divided by nine, it is impossible. Nevertheless, mathemagicians are an odd sort, very crafty, wily like the fox, devising new illusions all the time. They like to demonstrate that they can do the impossible. Some people even believe that they actually do what is impossible. That is a problem. — Metaphysician Undercover

The op uses an expression which represents an impossible division, 1/9, a ratio which cannot be expressed in numbers. There are different forms of divisions which cannot be expressed in numbers, some are called "irrational numbers". Since they are not numbers it was a mistake to start calling them numbers. But this is the bad influence which common vernacular has over logic, it inclines us to replace the rigorous logical definitions which are required for sound logic, with family resemblance. (It's similar to a number, so let's just called it a number, and then we can have a bunch of different types of numbers).

Yes, I think that's true. I'd go as far as to say that some philosophical discourse is deliberately engineered to serve this function.

In reality it is contradictory trivial to divide one into any parts, because then you are saying that it is not one, but however many parts you are dividing it into. — Metaphysician Undercover

To slice a pizza into equal slices, try a pizza cutter. Ten paragraphs of nonsense gibberish can be refuted with one kitchen appliance.

To most people, cutting one thing into multiple pieces is a triviality. Where's the contradiction you're describing? Cut one pizza into four equal slices, and you have four slices, each of which is one fourth of the pizza. What you're saying translates that there's a contradiction here because I'm saying that there are four slices and one pizza. How's that a contradiction? You would have to change the concept of "contradiction" to something that can actually be done, and the concept of "impossible" to something you can actually do, to make this argument stick, because we damned well can slice a pizza into four slices.

We can also cut each of those four slices in half, making eight slices. One of the four slices we had before is the same quantity of pizza as two of the eight slices we produce; i.e., it's a "particular quantity". It all works, MU. There's no contradictions except in your fictitious world where you can't use pizza cutters. In reality, we have pizza cutters and we can and do slice pizza.