An open thread, where at your weekend leisure, you can discuss anything you like.
Life, Culture and Politics from BrisVegas
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THE HUNGER GREAT IS UPON ME!
Behold, puny humans — I am Galactus, devourer of worlds and blog comments! I have come to your blog to satisfy my cosmic hunger, by devouring all who dare to… um… uh…
Hello? Anyone?
Man, maybe Australian hangovers really *do* last for three days straight… pretty impressive…
*sigh*
CoG @ 1,
My hangover’s still going from NYE and I didn’t even get drunk. (5-6 glasses of dry white, 2 of champagne.)Tho I do hear tell there’s this terrible stomach bug going around that makes you feel horribly sick, but not sick enough to spew. I know at least one person who has it, and I think I’ve got it as well.But maybe not. Maybe I just need to have some breakfast. (Other than various kinds of Xmas Cake. Living on Xmas Cake for nearly a week can’t be entirely good for you.)
Maybe there are lots of others like me who made a New Year’s resolution not to comment on blogs any mo…………oh, bugger.
Early Adversary gets the mescal worm, Galactus. Come around on about the 29th or so next year and we’ll try and fit you in.
You’d think that after Thursday’s Sunlander prang, that people wouldn’t be so foolish so soon.
Finally, from January 1, the Commonwealth of Australia considers my husband and I, married now for nearly 5 years (Toronto, ON, CA), a de facto couple rather than legal strangers.
I don’t know whether to be grateful or angry.
Combine the two, Quog, and be angrily grateful (or gratefully angry).
Paul Burns
If its the one Im seeing a bit of at work its a bugger. One girl has had it for 2 weeks and 2 more with it today. One of mine is spewing though. No energy, slight temperature, queazy stomach…
Nearly everyone here comes in and out of work on a bus so I expect to be seeing another one every day for about another week.
Sounds about right. years ago if I felt sick but couldn’t chuck, a glass of lemon saline used to clear the stomach. Its quite a low-level nastiness this bug. I’ve had it on and off since about a week before Xmas. Wasn’t brought on by a hangover either. Though, even with a reasonably small intake of alcohol my NYE one lasted a day and a bit. (Must get some lemon saline, see if it still works.)
The new Doctor Who has been announced!
I won’t risk spoiling people who don’t want to know here, but Catriona’s having a thread about it on Circulating Library. Suffice to say we were all surprised by the choice.
Nick Caldwell @ 10,
Apparently is screening ABC1 at the end of January. There were some mostly poor quality trailers on YouTube. Most of them looked like they were pirated and needed subtitles to be understood. Won’t spoil it but I love the period its set in. (Willing to be corrected if I have the date wrong.)
Ah, I’m actually referring to developments somewhat further in the future than the just-aired Christmas special.
Ah, well, I won’t ask for a spoiler. Something more to really look forward to.
Re Dr. Who. SBS revealed all on last night’s news. But it won’t be out till next year. Hint: check out Party Animals, Tuesday nights on ABC1. (It’s brilliant, btw.)
And … Colleen McCullough’s Thorn Birds is going to be made into a West End musical. I know it sounds like a joke, but apparently its true.
Wonder if Somewhere over the Rainbow or The Wonderful Wizard of Oz will be among the songs in it?
Oh dear, Quadrant hath been hoax-ed
http://www.quadrant.org.au/blogs/qed/2009/01/margaret-simons-and-an-apparent-hoax-on-quadrant
What a piddly little hoax. I expected something utterly brilliant like Earn Malley or Helen Demidenko. Ah, well, guess you can’t have everything.
Here’s more on it, with context. I didnt really get it either till I read this: http://www.crikey.com.au/Politics/20090106-How-Quadrant-swallowed-a-giant-hoax-.html
Here’s the money quote, from Crikey:
The Gould hoax is designed to be a companion and a counter to the famous Sokal hoax, in which the physicist Alan Sokal submitted a paper to a postmodern cultural studies journal to show that post modernists would “publish an article liberally salted with nonsense if (a) it sounded good and (b) it flattered the editors’ ideological preconceptions.”
The Sokal affair became part of the “science wars” which were a series of intellectual battles between post modernists and realists, and a companion to Australia’s “history wars”, in which Windschuttle has been a leading contender.
On the day Windschuttle informed “Gould” that the article would be published, the hoaxer wrote in Diary of a Hoax:
“For pity’s sake, Quadrant fell for my ham-fisted ruse! At least with the Sokal hoax, Alan Sokal was a bona fide physics professor. So it’s understandable that a journal editor might unquestioningly publish his nonsense. But so neatly did my essay conform with reactionary ideology that Quadrant, it seems, didn’t even check the putative author’s credentials. Nor it seems did they get the piece peer-reviewed. Nor did they check the “facts”; nor the footnotes. Nor were they alerted by the clues…Still, now my experiment has worked, I’m not sure how I feel about it. Do I feel schadenfreude? Not really. I feel ambivalent. I’m almost embarrassed for you, Windschuttle… I didn’t do this to be unkind to you personally. This experiment wasn’t designed with ill-intent, but to uncover hypocrisy in knowledge-claims, and also spark public debate about standards of truth when anything is claimed in the name of ’science’.”
continueing from here – http://larvatusprodeo.net/2009/01/06/windschuttle-sokaled/#comment-598992
Plotnitsky:
Marty:
Down & Out:
the definition that I have for a complex number is “Numbers that can be written as the sum or difference of a real number and an imaginary number.”
an imaginary numbers are defined as “Complex numbers with no real part, such as 5i.”
so there is real difference – complex number contain a real number and imaginary numbers do not.
a complex number can be written as a + bi, where ‘a’ is the “real part”, ‘b’ the whole number, and ‘i’ is the imaginary number such that i^2 = -1 (i squared is equal to negative one). e.g. ‘e + 5i‘ is a complex number, while ’5i‘ is an imaginary number.
now, neither of those numbers can be represented as the ratio of two whole numbers.
plotnitsky, again in part of the text that I elided, talks about ‘(real) irrational’ numbers and just plain ‘irrational’ numbers, to show how complex and (real)irrational numbers are related. among other things. he explicitly mentions Gauss at points, which really you have to if you want to discuss imaginary numbers.
another thing, this chapter is after the chapter on Bohr’s complementarity, which is very heavy going, in which Plotnitsky argues that in Bohr’s own interpretation of quantum physics, as revealed in his lectures and in correspondence with Einstein over at least two decades, it is impossible to speak of quantum ‘objects’ per se. what you have left are complementary sets of classical ‘effects’ on measuring instruments (or ‘phenomena’), and an actual theoretically unknowable reality underneath it (as analogous to Heisenberg’s principle of uncertainty). In other words, Bohr has a radically non-classical epistemology of quantum physics which admits the possibility of unknowable, i guess as a sort of absolute limit of knowledge itself.
the relationship between this and other ‘non-classical’ philosophy is explored in the remainder of the book.
Lacan, by the way is usually accused of relating the phallus to the imaginary number using a bastardised calculus. but, apart from having some basic misrepresentations, Plotnitsky attempts, and i believe succeeds, in showing that Lacan and others lumped in with him have a philosophical rigour about their writing that must be engaged with on their own terms, and no airly dismissed in some of “two cultures” baiting by people that Plotnitsky regards, I paraphrase here, ‘as wrong on Lacan as they accuse Lacan as being on mathematics, and in certain areas, even more wrong on mathematics as Lacan is’.
Plotnitsky, BTW, is Professor of English and Director of the Theory and Cultural Studies Program at Purdue University (English Romantic literature seems to be his main area), and he holds a MSc in Mathematics and is a member of the American Mathematical Society.
Errors above are likely to be errors of my representation, and not of Plotnitsky’s argument.
Hey, Tyro. welcome back, or welcome me back. whatever.
very keen to continue the discussion, but i’ve got another brain-intensive task right now. i’ll be out of action until late tonight, or maybe tomorrow. i’ll look carefully at what you’ve written and get back to you soon.
i didn’t want to write more than necessary on the other thread, but i will say i investigated plotnitsky a little after posting last night. and i read a little of his book on google books. he clearly has a decent mathematical background. (a masters in maths and membership of the ams don’t particularly impress me. and, i’m not convinced of exactly how mathematically literate he is. but there’s ample evidence that’s he’s no mathematical fool). but, that doesn’t mean that the passage you quoted is free from error (i think it clearly is not), or that his writing on mathematics in general is beyond criticism. in brief, what i read suggests to me that he knows what mathematics is, but doesn’t necessarily write about it carefully, nor say much interesting about it. but i acknowledge that this is a very tentative judgment, which may reduce to nitpicking.
will get back to you asap on your post above.
marty, i’ll try also to respond, i just had to spend a couple of brain-sapping hours on something else; and friday and saturday will be complete no-go zones for me .. saturday night will be getting plenty drunk too, so can’t promise any brain cells for sunday either.
Just a quick question – not having read the book – is the issue about Lacan’s use of mathematics? If so, some relevant info on where he derived his ideas from can be found in Elizabeth Roudinesco’s Jacques Lacan. Shouldn’t be hard to track down if you have access to a uni library. Years since I read it – so I won’t comment further, but obvs some people are quite interested!
That’s one of the issues that *Sokal* has with Lacan, yes.
Worth tracking down Roudinesco’s book, then! It’s a very comprehensive portrait. Mind you, the impression I was left with was that Lacan was a power hungry nutcase. Not that that invalidates his work, of course. Having said that, I’ve never had much time for his work. And I’m not familiar with Social Text’s back catalogue, but I don’t think that Lacan = postmodernism, if that’s what Sokal was saying. If you actually go into the history of that phase of French thought, he’s a bit sui generis. Deleuze and Guattari’s stuff, for instance, is certainly not Lacanian, and is in many ways a reaction against Lacan’s Freud.
I think the ‘take-away’ point here is that Plotnitsky is primarily writing philosophy, grounded in mathematics and quantum physics, rather than writing mathematics per se (and so maybe it falls in the genre of ‘Philosophy of Mathematics’). So of course there is never anything “beyond criticism”. Criticism, (or critical engagement) actually, almost being the point of the conversation, as it is.
mark and tyro (and d & o if you’re here), i think the very first question is not how lacan/plotnitsky/sokal-bricmont use mathematics, but how *mathematicians* use mathematics. specifically here, what do the terms “complex” and “imaginary” and “irrational” mean for mathematicians, and why? i’m (tentatively) happy to enter what i’ll probably regard as the swampland, but i’d first very much like to establish where the solid ground is. it’s not hard stuff, and it may in the end have little bearing on lacan’s/plotnitsky’s philosophizing. but i think it’s important. the use of technical terms must be understood and agreed upon before any re-use (or abuse).
i’ve been pondering this tonight, while doing my other work, and think i have a framework for writing about this. will probably post something in a couple hours.
The set of all real numbers is the continuum. do we need to talk about fields and the least upper bound? or is the continuum sufficient for our purposes?
Integers are the set of all positive and negative whole numbers, i.e. { … -3, -2, -1, 0, 1, 2, 3, … }
Rational numbers are those real numbers which can be expressed as a ratio of an integer with another non-zero integer. e.g. 13/17, 10/3, -47672/10^43, 0/7 (but note, not 7/0).
Irrational numbers are those real numbers which cannot be expressed as a ratio of of an integer with another non-zero integer, e.g. e, pi, the square root of 2, and so on.
I don’t think we’ll need algebraic and transcendentals.
Imaginary: i is the imaginary number such that i^2 = -1. All whole integer multiples of this number are also called imaginary numbers, e.g. 5i (sorry gonna dispense with the italic i at this point).
Complex: a complex number is an imaginary number with an additional real part, as I said above:
A Gaussian integer is a complex number of the form ‘a + bi’ for which both ‘a’ and ‘b’ are integers.
Looing around for a formal definition of transcendental numbers, I found that Wolfram (the Mathematica company) defines an Irrational number without reference to the real numbers as
And transcendental numbers are defined in part as “(possibly complex)”.
http://mathworld.wolfram.com/IrrationalNumber.html
Hi Tyro et al,
This’ll be the first of a number of posts (hopefully rational, probably 3), in attempt to keep separate the different points of contention.
Note, that i am not looking to say anything whatsoever about plotnitsky’s (or lacan’s) philosophy. that may well be the take-away point, and that may well mean that nothing i write here has any grand bearing. in particular, i wouldn’t touch plotnitsky’s, or anybody’s, philosophising on quantum mechanics with a barge pole.
My goal here is to set the groundwork for discussion of plotnitsky’s (and, if you wish, lacan’s) use of mathematical terms, and his criticism of sokal-bricmont’s use. later, i’m happy to address whatever mathematics you wish. but I think the current discussion is best broken down into 2 x 2 sub-debates:
a) The definitions of “complex” and “imaginary”
b) The definition of “irrational”
i) Mathematicians doing the defining
ii) Plotnitsky and/or lacan and/or bricmont-sokal doing the defining
The swamp is at (b)(ii), and so is best left for last.
now, to the extent that plotnitsky is in (ii), i have no fundamental problem. i may think his definitions weird or unorthodox, and i may have no clue to the philosophical value, but that’s his business.
however, to the extent that plotnitsky (or lacan) is conflating (i) and (ii), i do very much have an issue. and, as i shall try to make clear, i don’t currently know how else to interpret
the quoted criticism of sokal-bricmont.
So, my plan now is to cover (a) in the next post, and (b) in the next-next post. as I said, the real issues are with (b). i don’t think anything really hinges on (a). but it’s a good warm-up. and, in regard to (a), i think plotnitsky is sloppy to the point of error, and i want to make clear why.
hi tyro, happy to discuss all that with you. but really, there’s no need to sort out the current issue. interestingly, wikipedia and wolfram are not very helpful here.
o.k. let’s sort out the “complex” and “imaginary” stuff.
let’s take for granted we know what a real number is. hugely interesting mathematics, but i don’t see anybody disagreeing on that. so then:
1) there is no disagreement i see on what a COMPLEX NUMBER is (though see the very last section in this post). as you have written above, they’re things of the form
A + Bi
where A and B are real. (you seem to suggest above that B be a whole number: that’s an unusual constraint). again, we can argue about what i really is, but i don’t see anybody disagreeing here. (again, see the final remark). So, for example 3 + 2i is complex, and so is 2i ( = 0 + 2i ), and so is 3 ( = 3 +0i ).
(2) Now, this is part of the plotnitsky quote in relation to complex numbers:
“all … complex numbers are, by definition, irrational, since, not being real numbers …”
Well, if we’re still in (i), using mathematicians’ definitions, then this is simply false. 3 is both a complex number and a real number. (this was point (d) in my original response to the plotnitsky quote).
Now, as i said, i don’t think much hangs on this. i don’t really believe we’re in (ii). i think plotnitsky is simply being sloppy. there is also a chance that he is being sophistic (see the very last section in this post).
(3) next, “IMAGINARY NUMBER” is less consistent in its usage. the three ways it is commonly used are:
*) A complex number A + Bi
**) A complex number A + Bi with B non-zero
***) A complex number A + Bi with A = 0 (and maybe with B non-zero)
Note that 2i is imaginary by all three definitions, 3+2i is imaginary by the first two definitions, and 3 is imaginary by the first definition. (note also that without the parentheses then 0 would be imaginary by the third definition).
There’s no right or wrong here. Probably the most common usage of the term is as in (**): that is, an imaginary number is a complex number which is not real. usually (***), and usually without parentheses, is referred to as “purely imaginary”. your definition above seems to be along the lines of (***) (with parentheses?), though there is some suggestion that B must a whole number. I have never heard the expression used that way, though i guess it’s possible in certain fields of number theory (where gaussian integers would also enter). The use of (*) is common if one is referring to the whole set of complex numbers, as a world in which to do mathematics: here “imaginary numbers” is commonly used as a synonym for “complex numbers”.
(4) What does Plotnitsky mean by “imaginary numbers”?
“all imaginary … numbers are, by definition, irrational, since, not being real numbers …”
If Plotnitsky means (**), or (***) with parentheses then (ignoring “irrational” for now) this quote is fine. But note that with any definition, “imaginary numbers” is a subset of “complex numbers”. Thus, the expression “all imaginary and complex numbers” contains redundancy. This was point c) in my initial response to the quote.
5) Finally, is there a genuine mathematical sense in which all complex numbers are not-real? Yes, there is, as i’ll describe. If this is what plotnitsky means then he is not wrong: he is however boring and too cute by half.
If we try to get straight what a complex number *really* is, we have to come to terms with what i really is. there is no need for all the details: the upshot is that a complex number like 3 + 2i is (for example) defined to be an ordered pair (3,2). in this sense 3 is not a complex number: the complex number we think of as being 3 is really (3,0).
In the above sense, Plotnitsky’s quote can be regarded as correct (again ignoring the irrational bit). But, this is simply pedantry: it is not how any mathematician thinks about doing mathematics in the complex world. For any practising mathematician, the complex number (3,0) and the real number 3 are identified. (The technical term is “isomorphic”).
To put the issue another way, how do we view going from the world of real numbers to the world of complex numbers? Technically, we are creating a whole new world. But conceptually the way mathematicians think about it is to expand the world, with the world of reals a subset of the world of complex.
And just to make clear this issue, and the pointlessness of separate-world pedantics, ask yourself the following question: is 3 a rational number? The natural conceptual answer is “yes”. But, the technical answer is “no”, for the very same reason: to go from the world of whole numbers to the world of rational numbers, we have to technically create a whole new world. And once again, a rational number such as 3/7 would not be real: going from the rational world to the real world involves creating a whole new world.
Foundations of mathematics is fun, and important. (set theory, anyone?) But it does not indicate what mathematics is, what mathematics is to mathematicians.
Well, I presume that is more than you expected or desired. obviously more than enough for now, maybe forever.
The issue of plotnitsky’s use of “irrational” is quite separate, but i’ll at least wait until the complex/imaginary stuff is sorted before going onto (b). so, feel free to comment on or question the above, including “you gotta be kidding” if that level of detail is considered inappropriate here.
I don’t have time at present to really get into this. However just enough to say this;
As we are arguing philosophy (ultimately), what’s important is the foundations of number theory – not what “mathematicians think of mathematics”. In my view this latter thing reads like its anthropology or something like that. Although it definitely has bearing on epistemology of mathematics, not directly in this case.
So the field of complex numbers contains a subfield which is the set of real numbers. the set of integers is not a field. In my (limited) experience a complex number is typically defined as having a real and an integer part, i.e. (pi + 5i). I suppose it is possible for the imaginary component to consist of a real component and i, but I don’t fully accept that definition.
I produced a one formal definition of an irrational number that doesn’t insist that they are actually real numbers – just numbers that don’t have any whole number ratio – and what Mathematica thinks of the issue is important, and the definitions there are fully edited and referenced, unlike most other references.
So by this definition, any “complex number” which is not also in the subfield of the real numbers (which is to say a complex number with a non-zero imaginary component), IS is an irrational number, there are no integers which exist in ratio to the number.
Plotnitsky carefully represents that there are irrational numbers defined as real numbers the “(real) irrationals” and irrational numbers which are merely incapable of being defined as ratios of integers. And I can (as in, did) show this is not an necessarily uncommon formal definition of irrational numbers.
Philosophical argument, like mathematics, is typically proceeded by proposing axioms and then developing propositions which are shown to be true or false according to the given axioms, which develop into theorems, which can be used to define the truth of new propositions leading to new theorems, and so on. And in the axiomatic system thusly constructed as above the statement that “complex numbers with a non-zero imaginary component are also irrational numbers” is definitely a true proposition.
Does it matter if irrational numbers are all real or not? (except to propose a different axiomatic system in which complex numbers with a non-zero imaginary component are not irrational). I don’t really think there are nasty implications for number theory one way or the other. Mathematicians socio-technical conventions, maybe so, but then I don’t think we are concerned with that here.
Of course, this is about two or three paragraphs of Plotnitsky’s reasoning in one chapter of his book. The point is leading to “non-classical” philosophy of the type of Bohr’s, or Lacan’s, which utilises (indeed, relies on) “classical” philosophy in its formalism, but is not in totality that classical philosophy (if my understanding is not faulty, which I won’t claim at this juncture because it is pretty heavy going stuff).
On the other hand, we don’t want to be sent mad by it all, like poor old Georg Cantor.
I probably won’t be able to respond to anything else for several days, but i don’t think it’s been an unfruitful argument thus far.
No, it’s not anthropology, though I can see why you might think that. And it’s not a foundational question unless plotnitsky’s use of “complex number” is as in point 5. If it is, he’s not wrong (ignoring irrational), merely ridiculous. (more on this later).
I appreciate that what i’ve written is a lot to digest. And I’m not bothered if you don’t have time. I’ll look at your post and answer your questions shortly. i’ll also try to make clear why this is definitely not a case of colloquial mathematicians versus careful plotnitsky.
“definitely not a case of colloquial mathematicians versus careful plotnitsky.”
no because it is sloppy Sokal and Bricmont vs carful Plotnitsky !!!
possibly …
Well, it is *you* who is positing it Plotnitsky vs the mathematicians. When it is plainly obvious from his publication record and his professional standing that he can be regarded as one, if slightly unorthodox.
Also I just don’t accept the idea of what “mathematicians” say. They are not a monolithic block of people who have no disagreements, or areas of expertise and interest. That’s just as silly to me as saying what “physicists” think when clearly there are definitely large numbers of physicists who must be wrong if some other large numbers are right.
What is important, in a foundational sense, is what formal number theory can be shown to say about the matter, and then, what philosophical concepts can be extended from the particular axiomatic system thus developed.
In answer to your questions:
(1) The reason my post above came across as anthropological is because “imaginary number” is more of an undergraduate and a colloquial term. It does not have an accepted precise meaning (though all the meanings are closely related). I only used it because Plotnitsky used it.
(2) By comparison “complex number” does have a precise and accepted meaning, and is as I defined above. (this is modulo the foundational question, which i’ll come to). It is not relevant to our plotnitsky debate, but your understanding that the “i” bit must be a whole number is simply false. All mathematicians, all everybody, would agree that 3 + e*i is a complex number.
(3) “So the field of complex numbers contains a subfield which is the set of real numbers.”
Yes and no. This is the foundational question. First of all the fact that these are fields is true, but a red herring. Now the subtle bit:
(a) Imagine you start with the world of complex numbers. You can then ask, what natural subworlds are contained within? One such subworld is the world of real numbers. It is in this sense that your understanding is correct. This is the mathematican viewpoint.
(b) Imagine instead you start with the world of real numbers. You can then ask, what natural new larger worlds can i create? One such larger world is the world of complex numbers. HOWEVER, the technical procedure for creating the new world actually creates a whole new separate world: the original world of real numbers is NOT contained in the new world. This is the foundational/pedantic viewpoint. from this viewpoint, your quote is false.
( c) Now, it is possible that Plotnitsky is taking the (b) point of view. But, for reasons I tried quickly to indicate (and i can try again upon request), this would still not make sense of plotnitsky’s paragraph, EVEN from the foundational/pedantic point of view.
( d) “I produced a one formal definition of an irrational number that doesn’t insist that they are actually real numbers – just numbers that don’t have any whole number ratio.”
No, this is not a formal definition, unless you tell me what “number” means. Now, you and/or wolfram may simply be assuming that “number” means “complex number”, but that is your(s) assumption. There are other choices.
Almost certainly, Wolfram is assuming in that context that “number” means “real number”. I know of absolutely no one who is mathematically literate who would do otherwise, who would include 3+2i as an irrational number.
(4) “Does it matter if irrational numbers are all real or not?”
Yes and no. It certainly matters to know how mathematicians use the term, especially if you’re gonna criticise them. And to use the term otherwise is to invite a hell of a lot of confusion.
Is there value in the broader definition of “irrational”? Maybe, but for reasons I can indicate I am hugely skeptical. The mathematician definition, restricting it real numbers, is extremely natural.
“Also I just don’t accept the idea of what “mathematicians” say. They are not a monolithic block of people who have no disagreements, or areas of expertise and interest.”
On the overwhelming majority of definitional questions, yes they are. You don’t have to accept it, but it’s true. For mathematicians, 2+3i is not irrational, no matter what the hell Plotnitsky is, or what he says. He can define “irrational” however he wants, but that cannot change the formal, universally accepted (minus wankers) definition of “irrational number”.
The only caveat to that is the foundational question, the formal process of construction of mathematical objects. But this has nothing to do with whether irrationals are real.
“Well, it is *you* who is positing it Plotnitsky vs the mathematicians. When it is plainly obvious from his publication record and his professional standing that he can be regarded as one, if slightly unorthodox.”
No, as far as I can tell Plotnitsky is not a mathematician. 1) Anyone can become a member of the AMS (and good thing, too): membership doesn’t make them a mathematician. 2) Plotnitsky has a masters in mathematics. This may have had an original research component, but it is not clear, and many (arguably most) masters in mathematics do not. 3) As far as I can tell from his publication record, Plotnitsky has published no research in mathematics: i.e. he has never proved a new theorem. At closest, Plotnitsky seems (arguably) to be a philosopher of mathematics.
I’m not trying to be elitist, nor to denigrate Plotnitsky. One can certainly haggle over the definition of “mathematician”, and there is nothing belittling about being a philosopher of mathematics. But they’re not the same thing.
But in any case, even if Plotnitsky is a mathematician, it makes no difference. If one unorthodox but genuine biologist decides that animals which go “woof!” are also cats, it doesn’t change the meaning of “cat”.
Marty, this will be my final post on this topic, as I have much better things to do with my time that pointless arguments over philosophical reasoning.
First, there *is* a definition of the imaginary number. It is i. Defined such that i^2 = -1 – you cannot just hand-wave that away “3 + ei” is complex. Of course it is. But not until i is the imaginary number.
You haven’t tackled at all what I said about axiomatic systems and the development of an argument from axioms and how that might relate to what Plotnitsky writes. So I just don’t see there’s anything we can talk about.
It is the foundation for all philosophical reasoning and I can’t see there’s any progress in dialogue without it. What “mathematicians” think is of secondary or even tertiary importance, it’s not central to anything Plotnitsky says or discusses, it’s just a canard that you have introduced.
Especially irrelevant is the line ” It certainly matters to know how mathematicians use the term, especially if you’re gonna criticise them.” because no one is criticising mathematicians. It’s just a straw man of your own making. I can see no point in argumentation around it because it is pointless.
Sokal and Bricmont are stupendous dunderheads. They do not understand philosophy. And they they critique something for innaccuracies from a perspective that can be said to contain flaws of the very nature that they castigate their target for.
End of argument as far I am concerned. Their work is of no philosophical weight.
tyro, you had an opportunity to learn here. i was offering you something good. i’m sorry you have failed to realise that.
marty, that’s totally arrogant to act as if a conversation is not two way, i.e. that it’s my knowledge which is faulty, while yours is perfect and without blemish. after all, i’m the one here who has actually read the texts in question.
tyro, i’m sorry if it came across as arrogant. but the fact is, i know a lot about mathematics, and mathematical foundations and axiomatization, the whole game. i didn’t presume to know anything about plotnitsky, and i wasn’t trying to say anything about plotnitsky, except his use or re-use of mathematical terms. and even there i was very careful to consider different reasonable interpretations.
you seem to care about plotnitsky. you also seem to know very little mathematics, which at least at some level seems important to understand plotnitsky. i was trying to help you.
i am open to the idea that plotnitsky is using or re-using mathematical terms reasonably, though i haven’t seen it. on the other hand, you seem to simply presume it. you seem not to be open to plotnitsky being silly or wrong, at least on this one small point.
I was trying very hard to listen to you, to make sense of what you were writing about mathematics, and with genuine concern was trying to correct your misunderstandings. but you seem not to have listened to me at all: you’ve simply been taking sides. such are science wars begun.
marty, of course plotnitsky might be wrong – not just on some minor detail, maybe even horrendously categorically: my argument would be that “wrongness” of plotnitsky’s method isn’t in that particular detail.
I’m keen to defend him because i’ve read the texts in question and can see his point. I would ask you if you even know what that point is.
As for my ‘mathematical’ knowledge, again please stop being so insulting and arrogant. I’m no professional mathematician but I do have computer science – academically (many years before those things were turned in “IT” degrees) – and professionally, and before that some electrical engineering, it was only after that I did philosophy, and then became a history post-grad student.
I could say much the same about your apparent lack of demonstrated philosophical knowledge, given that epistemology is actually what is being debated here, not mathematics per se.
To be discussing “What mathematicians think” sounds to me like structuralism. Normally I would describe myself as a structuralist (I certainly am in terms of culture and history), but actually, I don’t think that logic is structuralist in the sense of its ultimate form – maybe how is come be understood amongst mathematicians is important. But I would regard for example, Russell’s theory of types to be quite independent of a community that reasons about it, at a fundamental level, because it’s not a rule-governed social system of signs at it’s core, even if it’s expressed within a particular community as such a set of social conventions.
I’m also well aware that mathematicians such as Hilbert and Godel were at opposite ends of the spectrum in their conceptions as to how mathematical epistemology ultimately arises and then proceeds. What about non-standard analysis? I know it is vigourously debated, or at least has been within living memory. These two things, among others, lead me to the conclusion that mathematicians don’t all necessarily think alike and that “all mathematicians” don’t all agree.
Because the only point of fundamental disagreement I see between us is over the irrational numbers (and I have already pointed out that Plotnitsky in fact is quite careful to distinguish between these two definitions). And you assert your correctness by an appeal to the authority of mathematicians. I say that is a logical fallacy.
All you are saying as far as I see so far is that there is a standard convention in mathematics that hold all irrational numbers to be also real. I don’t, in fact cannot, argue with that. Is there a consistent axiomatic system in which irrational numbers aren’t all real numbers? You haven’t shown that it isn’t consistent. Even though it could well be, an appeal to authority is simply not sufficient.
Tyro,
1) i know nothing of your mathematical background. i do know that above you have written a number of things about mathematics which are very confused and/or incorrect.
i was not in the least trying to put you down for this. But it seems clear to me that I know mathematics much better than you, and the mathematics itself seemed relevant. does that make me arrogant? if so, i’ll wear it. honestly, i was just trying to help you.
2) though i know about metamathematics and the logical and set-theoretic foundations of mathematics, i do not claim for a minute to have any expertise in philosophy, either in general or of mathematics. and i expressly disavowed making any comment on anyone else’s philosophy.
3) you are right that mathematicians don’t agree on everything, and certainly not on foundational and philosophical questions. all i said was that they don’t tend to disagree on definitions.
4) I wasn’t asserting that plotnitsky’s definition of “irrational number” was incorrect because it disagreed with the mathematical definition. here is what i was and am saying:
a) first, i was trying to make clear the mathematical definition of “irrational number” and make clear that within mathematics that definition is, for all intents and purposes, universally accepted.
b) secondly, i am saying that if plotnisky now wants to use the expression “irrational number” in a different manner then he is inviting confusion. at minimum, it seems reasonable to expect him to make clear that his use is different. and, one wonders why he doesn’t just use a different expression: why not simply call them Plotnitsky Numbers?
c) thirdly, IF plotnitsky is criticising sokal-bricmont for using “irrational number” in the mathematician sense, this seems like a very odd objection. (i have no idea and have made no comment on whether sokal-bricmont have been fair to lacan).
d) fourthly, a minor point, i criticised plotnitsky’s use of the expression “imaginary and complex numbers”. now, perhaps he says something elsewhere to clarify his use of those expressions. but, as it stands, i stand by my criticism.
5) i don’t know if “axiom system” is an accurate expression, but i’ll agree that i don’t see any internal inconsistency in what you say is plotnitsky’s definition of “irrational numbers”. HOWEVER, i did suggest, without explanation, that there are INTERNAL problems with his definition. that is, even for plotnitsky, i think his definition is problematic. If you wish, I can explain my concerns.
i see your point 4.b), but he does make a clarification, which i’ve repeatedly pointed out. point 4 in its totality i can accept from your position is a flaw in his argument, at that particular point. I think the case hinges on c) – I don’t know though if its accurate to say that he’s necessarily critqueing sokals pov from the “mathematician viewpoint” (that might be a flawed interpretation of mine, or, flawed interpretation of yours based on my incomplete or inadequate understanding or description).
point 2) … well you may not think so but you’re already commenting as such.
on another point – i don’t accept your claims of evaluation of my own mathematical skill. i’m not a ‘professional’ mathematician by any means of course, but i don’t believe i’ve made significant errors of logical expression in terms of mathematical philosophy. it is naive perhaps. now maybe that differs from the perspective of ‘mathematician mathematics’, but then it’s a question of being able to communicate across those gaps, which means you have to grasp my perspective as much as I must gather yours.
in some ways, that’s a fairly basic statement of what is wrong with sokal and bricmont. they assume that Lacan (et al) “misunderstand” mathematics, as from one perspective, that purely of the working mathematician, where statements about definitional terms cannot be stretched to meet other purposes. but there’s another perspective (and actually a multiplicity of them) where this could be perfectly good philosophical reasoning (but not mathematical reasoning, of course). but then, they proceed to monumentally misunderstand what’s happening in totality, and draw all sorts of idiotic conclusions about what is being argued. really. no-one denies that physical reality exists. i’ve never met such a person – not even the most committed and radical post-structuralist.
as way of analogy as a both computer scientist and philosopher i got awfully upset when I encountered the information theory people using “ontology” to describe … well i’m still not exactly sure what they mean by it? sets of information related by common concepts? anyway it gave me the shits. but after a while i accepted their use of it, and practically i have to deal with it from a practical perspective. i can discriminate that field’s use ‘ontology’, from say Bachelard, or W.V. Quine (actually i suspect it is Quine’s ontologic arguments that the information theory ontology is derived from, but maybe not).
here’s a little though experiment. say you were philosophising about ‘rationality’. might not the fact that mathematicians, and philosophers, sort some classes numbers into the categories of ‘rational’ and ‘irrational’ be of interest to your philosophic reasoning? depending on exactly what you are doing, it might not be that you have to particularly careful of exactly what the categories contain – just that those *labels* where used, by mathematicians and philosophers, to describe numbers. ok that’s not a sophisticated example, but i hope you get my point.
now you might not want to grapple with that, and stay inside just the world of “mathematician mathematics” but really if you want to deconstruct continental philosophy – even using the perspective of such mathematics, it has to be grasped, otherwise the critique is simply unreasonable.
otherwise i’m too hungover to engage further today, that’s exhausted me enough and i need vast quantities of coffee.
Hi Tyro,
Amazingly, I think we’re finding some common ground.
First of all, i couldn’t use “ontology” or “epistomology” in a sentence. Well, maybe “There goes a yellow ontology” – how was that? I wasn’t meaning at all to criticise your (or Plotnitsky’s) philosophy of maths. My criticism of your maths was only meant to be your interpretation of the internal mathematical meaning of mathematical expressions. I do have thoughts about the external philosophical meaning, and the existence of mathematical objects, but I’m definitely no expert and was deliberately trying to delay such discussion.
I definitely agree that the splitting of the world into “rational” and “irrational”, and “real” and “imaginary” is an interesting one. The little I read of Plotnitsky (google books isn’t easy), i could see that that was the kind of thing he was on about. So, for him it seemed the class of (mathematical) things which are not-rational is a natural class. But, as you seem to agree, re-using established mathematical terms is perhaps not the wisest way to go about this.
Having said that, i do have doubts about the use of plotnisky’s concept of (let us say) not-rational: my point 5 above. But, that can wait.
enjoy your hangover and cure.