[Interview] SPUNKY IN YOUR FACE
besides... even if you change modulo or anything... many of the concepts that relate to mathematics still remain immutable. for instance, the idea of '2' as in a set with two elements (whether we want to associate the symbol '2' with the idea of twoness is irrelevant) is still there. that doesnt change regardless of what people do. you maye or may not need to use it, but that doesn't mean it just disappears or becomes arbitrarily re-defined
Except then we run into some serious problems when (1) something is true under infinitely different (non-isomorphic -- i.e., non-equivalent) structures and (2) what if certain statements are neither true nor false under a given structure? Godel proved for first-order logic, any formal language that can express arithmetic (prime factorization, really), there are things that are true in the language (have models), but we cannot prove them within that model! For instance, the general continuum hypothesis (what is the size of the continuum?) is independent of the ZFC axioms of ZFC set theory. Is the sentence that says the continuum is the first uncountable infinity a statement that is true or false? If we depend on logic, this sentence is independent of certain structures, modeled in others, and false elsewhere! Not to mention the logic that really ties all this together is not set in stone. There are many types of logic (besides first-order predicate logic) that we can turn to: modal, paraconsistent, substructural, many-valued, fuzzy, and more!
You see, your step, to me, looks like you want to say "it's still true once we replace everything under the right definitions," basically. But then the truth of the statement is far from immutable. The statement is constant, the interpretations are mutable and the truth is contingent on those interpretations; hence, the truth is mutable!
Is truth a property of sentence? A relationship of sentences and models? How is that relationship defined? Logic? Is truth semantic (relies on meaning) or verificationist (relies on proof)? Whichever side of this multi-faceted coin you land on, there is mutability!
Well, the one escape is to say the Truths (big T!) of mathematics are not found in our truth-theoretic or proof-theoretic mechanics of logic and math, but then that sort of begs the question that there's a Platonic realm of mathematical truths (Forms) and how we ever know if our mathematics and logic ever "get it right." That position, mind you, is one a good portion of mathematicians (and philosophers) adopt at some level, though. The problem is to tie our actual mathematical practices requires we have some way of verifying our results, like we have perception of the real world to validate our empirical measurements (at some level). Godel, among many, posit strange abilities of intuition to "see" the Truth, but that just seems like baseless conjecture to me! But I digress
You see, your step, to me, looks like you want to say "it's still true once we replace everything under the right definitions," basically. But then the truth of the statement is far from immutable. The statement is constant, the interpretations are mutable and the truth is contingent on those interpretations; hence, the truth is mutable!
Is truth a property of sentence? A relationship of sentences and models? How is that relationship defined? Logic? Is truth semantic (relies on meaning) or verificationist (relies on proof)? Whichever side of this multi-faceted coin you land on, there is mutability!
Well, the one escape is to say the Truths (big T!) of mathematics are not found in our truth-theoretic or proof-theoretic mechanics of logic and math, but then that sort of begs the question that there's a Platonic realm of mathematical truths (Forms) and how we ever know if our mathematics and logic ever "get it right." That position, mind you, is one a good portion of mathematicians (and philosophers) adopt at some level, though. The problem is to tie our actual mathematical practices requires we have some way of verifying our results, like we have perception of the real world to validate our empirical measurements (at some level). Godel, among many, posit strange abilities of intuition to "see" the Truth, but that just seems like baseless conjecture to me! But I digress
Now children - this is a thread for asking spunky questions. Take your thought provoking mathematical arguments to a different thread. On that note:
1) What is your favorite food?
2) If you had to move to the US which state would you prefer to live in?
3) And the most important question of all: In the inevitable battle between good and evil (bots and raptors) - which side will you choose?
1) What is your favorite food?
2) If you had to move to the US which state would you prefer to live in?
3) And the most important question of all: In the inevitable battle between good and evil (bots and raptors) - which side will you choose?
Question: What do you think of this?
In "What numbers could not be" (taking as the initial argument for mathematical structuralism), Paul Benacerraf argues exactly the notion of number as sets. There are two completely usable definitions of numbers, as we typically define them to be associated with size (numbers are cardinal). For instance, the number 0 is just the empty set {}. The number 1 is the set {{}} = {0}, the set containing the empty set. The set 2 is just {{}, {{}}} = {0, 1}. In this fashion, all cardinals are sets containing the sequence of their predecessors. However, it all works out if we do it this way 0 = {}, 1 = {{}} = {1}, 2 = {{{}}} = {1}, 3 = {2}, ..., etc. In this way, the size of the set is always singular!
As Benacerraf argues, whatever we understand these mathematical critters to be, they most certainly cannot be entities unto themselves. Decades later, structuralists have basically established a good foundation that even the concept of number is "a position within a structure." To be an entity is to be an entity of an abstraction. Truth itself, as a property among these mathematical entities, is not individual (a property of an object) but holistic (a property of the parts to the whole). No doubt truth is fixed within the structure, but that's just to say "when I define everything my way, it works out how I say it does." The real question remains, what does it mean for a mathematical sentence to be true? Is it because we decided to pick a certain structure over another one? Is there a "right" choice of model (interpretation) among the infinitely available? These are significant consequences of structuralism because it not only has implications to the semantics (and truth) but to the syntax (how we describe) of mathematics. Some question, for instance, if ZFC is the "proper" model for mathematics at all. If the continuum hypothesis is not true under ZFC or false under ZFC, what is the proper extension to ZFC? (i.e., what axioms can we add?).
So it's not an issue of arbitrariness so much as its a problem with the very justification for the theories of mathematics we have developed. Most of it is a historical accident that it progressed as it has (just like in any science). But people like to think that mathematics uses logic, and logic has definitive deductive truth, so there is only "one right answer," so to speak. That isn't the case, though! As it turns out, our choice of one model was never chosen. It just happened to be the one that we latched onto. How do we know if it is "right?" In the history of physics, there were many bad theories (EM as a mechanical ether, e.g.), but we have ways to ultimately validate that--viz., a reality that it will run counter to! I use the ether example because in discussions of structural realism, what maintained after early theories of a mechanical ether disappeared? Only the interpretation! The mathematics in this unique case remained exact. The "structure," as it were, carried over to the EM theory we use today that doesn't involve a mechanical substance to carry the vibrations in EM fields. We simply posited a new interpretation on top of the mathematical structure that was preserved--viz., that there are EM fields. http://onlinelibrary.wiley.com/doi/10.1111/j.1746-8361.1989.tb00933.x/abstract
In "What numbers could not be" (taking as the initial argument for mathematical structuralism), Paul Benacerraf argues exactly the notion of number as sets. There are two completely usable definitions of numbers, as we typically define them to be associated with size (numbers are cardinal). For instance, the number 0 is just the empty set {}. The number 1 is the set {{}} = {0}, the set containing the empty set. The set 2 is just {{}, {{}}} = {0, 1}. In this fashion, all cardinals are sets containing the sequence of their predecessors. However, it all works out if we do it this way 0 = {}, 1 = {{}} = {1}, 2 = {{{}}} = {1}, 3 = {2}, ..., etc. In this way, the size of the set is always singular!
As Benacerraf argues, whatever we understand these mathematical critters to be, they most certainly cannot be entities unto themselves. Decades later, structuralists have basically established a good foundation that even the concept of number is "a position within a structure." To be an entity is to be an entity of an abstraction. Truth itself, as a property among these mathematical entities, is not individual (a property of an object) but holistic (a property of the parts to the whole). No doubt truth is fixed within the structure, but that's just to say "when I define everything my way, it works out how I say it does." The real question remains, what does it mean for a mathematical sentence to be true? Is it because we decided to pick a certain structure over another one? Is there a "right" choice of model (interpretation) among the infinitely available? These are significant consequences of structuralism because it not only has implications to the semantics (and truth) but to the syntax (how we describe) of mathematics. Some question, for instance, if ZFC is the "proper" model for mathematics at all. If the continuum hypothesis is not true under ZFC or false under ZFC, what is the proper extension to ZFC? (i.e., what axioms can we add?).
So it's not an issue of arbitrariness so much as its a problem with the very justification for the theories of mathematics we have developed. Most of it is a historical accident that it progressed as it has (just like in any science). But people like to think that mathematics uses logic, and logic has definitive deductive truth, so there is only "one right answer," so to speak. That isn't the case, though! As it turns out, our choice of one model was never chosen. It just happened to be the one that we latched onto. How do we know if it is "right?" In the history of physics, there were many bad theories (EM as a mechanical ether, e.g.), but we have ways to ultimately validate that--viz., a reality that it will run counter to! I use the ether example because in discussions of structural realism, what maintained after early theories of a mechanical ether disappeared? Only the interpretation! The mathematics in this unique case remained exact. The "structure," as it were, carried over to the EM theory we use today that doesn't involve a mechanical substance to carry the vibrations in EM fields. We simply posited a new interpretation on top of the mathematical structure that was preserved--viz., that there are EM fields. http://onlinelibrary.wiley.com/doi/10.1111/j.1746-8361.1989.tb00933.x/abstract
i firmly abide by the theory that the world can survive on rice and potatoes, my 2 favourite foods!
Washington State or Oregon. i'm a sucker for the pacific northwest... besides, i despise the sun with all my black, little heart. i take great pleasure on cold and darkness
i've always suspected that, secretly, all of us social science folks are raptors so i guess you now know where my allegiance is...
Washington State or Oregon. i'm a sucker for the pacific northwest... besides, i despise the sun with all my black, little heart. i take great pleasure on cold and darkness
i've always suspected that, secretly, all of us social science folks are raptors so i guess you now know where my allegiance is...
Hi Spunky
I had always been interested in interviewing you!
I had always been interested in interviewing you!
i don't mind, really. when i'm not in talkstats i'm working and then back here. my husband's on a business trip so i really have very little time to even leave the apartment, lol.
i'm 26... which makes me wonder whether or not that makes me the youngest talkstats regular contributor
that's a.... peculiar question because, as weird as it sounds, i don't have a favourite band/singer/genre. i like *songs*, usually only one or two songs of the same artist whereas i can't stand the rest of them.
for instance, from adele i like "rolling in the deep" and "set fire to the rain" (but nothing else)
from coldplay i only like "talk"
from lady gaga i only like "paparazzi" and "judas"
and the list goes on... heh. probably the only one genre i really like is techno/trance because i used to go to (somewhat illegal) raves very often here in vancouver. oh! and holy music like gregorian chants or mantras. i really like repetitive, soft sounds.
i'm currently hooked on this when i work:
http://www.youtube.com/watch?v=Dlr90NLDp-0
yes
piano!
well, i really liked egypt when i went there because when i was like in 4th grade i became 'OBSESSED' with ancient egypt (everything from its inception till it was conquered by the roman empire) so my mother told me that if i continued on with said obsession, she would take me to egypt after grade 6th (which in mexico is when you transfer from elementary to middle school). so the obsession continued and she took me there. and everything was SO FREAKIN MONUMENTAL. i remember crying when i saw the pyramids because i had only seen them in books and on the tv until that moment :')
asia in general also impressed my widely, but mostly thailand. with the exception of thailand i've only been in big asian cities (kuala lumpur, singapore, sentosa island, seoul, etc.) or resort areas (penang island, etc.) and i've found them quite westernalised. so when i was hanging around there i remember telling my hubby "don't you feel like we're in miami or something like that?". but in thailand i've spent enough time to kind of explore the little corners of bangkok, visit less touristy areas and i'm amazed at just how nice thai people are. so kind and warm-hearted...
and i also have to mention venice because as a young, gothic, gay teen the only gay stuff i came in contact with was what came out in the vampire novels of anne rice (interview with the vampire, anyone?). and the first book i read (which was HEAVY on the homoerotic stuff) was "the vampire armand" which takes place, for the most part, in 1500s Italy. so that created an idealized, romanticized verion of venice for me and still holds a special place in my heart.
before i watched 'beautiful mind' and made the complete switch to math, i was actually very interested in psychology. but not just any branch of psychology, psychoanalysis more specifically. i really wanted to become a hard-core Freudian analyst and even wanted to become a psychiatrist just so i would be allowed to train in Vienna as a Freudian analyst. during my teen years i was really into Jung and his ideas of synchronicity and the collective unconscious... but i was also into big foot, UFOs and tarot cards so... heh. i think to be an analytical psychologist (the school founded by jung) you need a very specific kind of patient who's capable of powerful insight with relative ease. not many people can do that so i think most of Jung theories are interesting stuff to know but not very useful as clinical practice. unless you're carl jung, of course, who seemed to have the uncanny ability to just detect problems on people.
so... Freud. although nowadays i have to say if i wanted to take the route of psychiatry i would want to be come a biopsychologist or neuropsychologist because i think that within those areas lies the secret to understanding the human mind.
final fantasy 7 (the original one for play station 1) and kingdom hearts 1 (for play station 2). you have no idea how many fanfictions i read, fanart saved and fanforums i had membership to for those games.
"A beautiful Mind" and "\(\pi\), the order of chaos".... although "Cloud Atlas" is moving very quickly towards becoming top #3, particularly given how the world is today. i mean, just have a look at this part:
http://www.youtube.com/watch?v=RYLpQFuxV5Y
at min 3:35
"If I had remained invisible, the truth would have remained hidden, and I couldn't allow that.”
i like the Wachowski's work (the Matrix, Cloud Atlas,), Daniel Aronofsky (Requiem for a Dream, Pi, The Fountain), Tim Burton (Nightmare Before Christmas), and a little known, young French-Canadian director by the name of Xavier Dolan
actors/actresses... i think i'm only somewhat fond of Johnny Depp and Meryl Streep but i dont follow actor's lives very much. oh! and Tom Hiddleston for his work as Loki in the Avengers!
just vegetables. i was a strict vegan from say age 6 or 7 until my late teens so meat (of any kind) and i have never been the best of friends...lol
Andrey Nikolaevich Kolmogorov is high up there on the list not only for his brilliancy but because i think it reminds people of just how much the statistical sciences could have advanced had the soviet Union allowed scientists to communicate with each other freely. also because of the rumours he had an affair witha nother prominent russian mathematician, pavel alexandrov
C R Rao because he has touched so many areas of Statistics that i think we've all encountered him in one way or another or used some statistic or result developed by him. for instance, i know he's responsible for discovering a specific set of transformations of Wilk's lambda that get's reported in SPSS everyime someone does a MANOVA
Nicholas Metropolis who helped develop the ideas behind the Metropolis-Hastings algorithm and anyone who does Bayesian estimation is thankful for it
and these last two are just from my area in the social sciences:
Karl Gustav Jöreskog who, in my opinion, almost single-handedly saved the field of Structural Equation Modelling because not only did he provided some of the much-needed mathematical formalism behind the use of Factor Analysis, but also came up with the likelihood equations AND LISREL software package for estimating parameters using them.
Frederick Lord & Melvin Novick because in their seminal book "Statistical theories of mental test scores" they helped set psychometrics on firm mathematical basis. without that book, i think most of what people like me or Lazar do would just be smart number trickery, devoided from any statistical foundations.
well... you're lucky. at least you got to see it live! all we get here are crappy pseudo new rock bands. the west coast can suck sometimes when it comes to culture...
uhmm...
http://www.youtube.com/watch?v=QH2-TGUlwu4
uhmm...
http://www.youtube.com/watch?v=QH2-TGUlwu4