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| Philosophies of math...Includes essay...seriously
Posted: 11/4/2009 9:10:41 PM |
Is it my drinking or do I have B.O.?
It is definitely your B.O. 
It could be said that a language is a system of communication that communicates Ideas and concepts for a desired effect or purpose. As such to understand mathematics we must understand it in the same way we understand the parts of speech and syntax of a sentence. The difference is that math is extremely rigorous and almost immune to false logic
And within the language of math, one cannot use rhetoric...in its purist form...because math is immune to false logic... | |
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| Philosophies of math...Includes essay...seriously
Posted: 11/5/2009 1:07:26 AM |
math is immune to false logic...
...but the Bible is not. Much as I hate to introduce religion to the discussion, I can't resist noting that the three statements: "God is omniscient", "the Bible is the inspired word of God", and "God never lies" CANNOT all be true, and the Bible proves it. I'd provide the proof now, but it might be more fun to see if anyone who thinks all three can be true can make their case first.
In the meantime, I'd like to point out that even now, math itself is uncomfortable with "completed" infinities. They resulted in a split in mathematics that continues to this day. The Intuitionists (that I have been calling constructivists) would assert that the Platonists are just a bunch of religious fools if they believe the "law of trichotomy" (which states that every number will be either positive, negative or zero)...Why? because they would say people like Gödel used a false logic in assuming the existence of mathematical objects like completed infinities (which drove him to produce his incompleteness theorems). They would say that mathematics does not exist except as it is created and one could never construct something that required an infinite number of steps; it therefore can't exist.
Turning to the prior analogy between English and mathematics, the Platonist would argue that English exists as a complete language (i.e. all English words existed before there was anyone to speak them), where the intuitionist/constructivist would argue that English words are created in the mind of Man and don't exist before they are coined. | |
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| Philosophies of math...Includes essay...seriously
Posted: 11/5/2009 2:40:11 AM | I am finding that most of the time it is instead ignorance, short-sightedness, or selfishness leading to misplaced loyalties, that appear as false logic. Mostly because I've yielded to it in spades as I am growing up. I reckon, I reckon.
I could travel to the end of time. Check to see if the original propositions which laid the foundation for math, were not later proven to be short-sighted and require some amendment. The universe could give us a ring-a-ding-ding one day and say, "Hey, I see y'all are fixated on 1, 2, 3 and schtuff like that. I really appear that tangible to you? I'll try harder to reveal something that cuts through the tendency to consolidate before its time."
But I broke my time machine, or perhaps it really was a clothes dryer.
I seriously need to fix that thing and find out.  | |
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| Philosophies of math...Includes essay...seriously
Posted: 11/5/2009 7:47:07 AM |
In the meantime, I'd like to point out that even now, math itself is uncomfortable with "completed" infinities.
That seems to be an interesting problem that has far reaching implications for everyone. For the platonist, I wonder if there is an infinite amount of words to be spoken in the english? | |
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| Philosophies of math...Includes essay...seriously
Posted: 11/5/2009 4:18:31 PM | RE Msg: 28 by JustDukky:
math is immune to false logic... ...but the Bible is not. Much as I hate to introduce religion to the discussion, I can't resist noting that the three statements: "God is omniscient", "the Bible is the inspired word of God", and "God never lies" CANNOT all be true, and the Bible proves it. I'd provide the proof now, but it might be more fun to see if anyone who thinks all three can be true can make their case first.
In the meantime, I'd like to point out that even now, math itself is uncomfortable with "completed" infinities. They resulted in a split in mathematics that continues to this day. The Intuitionists (that I have been calling constructivists) would assert that the Platonists are just a bunch of religious fools if they believe the "law of trichotomy" (which states that every number will be either positive, negative or zero)...Why? because they would say people like Gödel used a false logic in assuming the existence of mathematical objects like completed infinities (which drove him to produce his incompleteness theorems). They would say that mathematics does not exist except as it is created and one could never construct something that required an infinite number of steps; it therefore can't exist. That's a difficult problem, because the concept that all numbers are either +ve, -ve, or 0, is called a "tautology" when I was taught degree mathematics, that being, a statement that has to be true, and that is because of the law of the excluded middle, that something cannot be true or false. You have a necessary conflict, between the fundamentals of decisive logic, and the proof that multiple infinities exist, that lead to the existence of a "set of all sets". However, I don't see that as a problem with "completed infinities", because you still have the same conflicts, even if you believe only that numbers "tend" to infinity, but never get there.
Turning to the prior analogy between English and mathematics, the Platonist would argue that English exists as a complete language (i.e. all English words existed before there was anyone to speak them), where the intuitionist/constructivist would argue that English words are created in the mind of Man and don't exist before they are coined. That sounds suspiciously like "essence precedes existence" versus "existence precedes essence", or empiricism versus existentialism. Life doesn't HAVE to fit into those dichotomous views either. | |
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| Philosophies of math...Includes essay...seriously
Posted: 11/5/2009 7:44:27 PM |
However, I don't see that as a problem with "completed infinities", because you still have the same conflicts, even if you believe only that numbers "tend" to infinity, but never get there.
The point is that there is a fundamental philosophical difference. The constructivist doesn't believe completed infinities exist because they cannot be constructed. All they will allow to exist in their mathematical world is finitary constuctions which are actually made.
That sounds suspiciously like "essence precedes existence" versus "existence precedes essence"
Is a pile of bricks and a set of plans a house, or does the house not exist until it is built? THAT is the essence of the philosophical difference between the Platonists and the constuctivists. | |
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| Philosophies of math...Includes essay...seriously
Posted: 11/6/2009 8:25:32 AM | RE Msg: 32 by JustDukky:
However, I don't see that as a problem with "completed infinities", because you still have the same conflicts, even if you believe only that numbers "tend" to infinity, but never get there. The point is that there is a fundamental philosophical difference. The constructivist doesn't believe completed infinities exist because they cannot be constructed. All they will allow to exist in their mathematical world is finitary constuctions which are actually made.
That sounds suspiciously like "essence precedes existence" versus "existence precedes essence" Is a pile of bricks and a set of plans a house, or does the house not exist until it is built? THAT is the essence of the philosophical difference between the Platonists and the constuctivists.
I lean towards constructivism. But I am not so restrictive as to be a "finite constructivist", and I am not so restrictive as to be purely Platonic. | |
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| Philosophies of math...Includes essay...seriously
Posted: 11/6/2009 10:05:18 AM | we are left with the fundamental problem that infiinity has as much validity as irrational numbers. Yet, everyone seems to accept that irrational numbers exist.
Who is "everyone"? An intuitionist would not assume the a priori "existence" of a number that cannot be exactly specified or proved. For instance, Brouwer showed that an exact value of "Pi" cannot be constructed, so in what sense does "Pi" exist for the constructivist/intuitionist?
Classical mathematics "proves" the existence of (for instance) the square root of two, but intuitionist mathematics does not assume it exists because it "must". The square root of two must be constructively proved without using the law of the excluded middle. (i.e. it is not sufficient to say it can't "not exist" therefore it must exist)/
As Kronecker once said to Lindemann:
"Of what use is your beautiful investigation regarding Pi? Why study such
problems, since irrational numbers are non-existent?"
So I can think of at least one mathematician you'd have to exclude from the set called "everybody."
Once you have the bricks and the plans, it is in the process of being built. However, until it is finished, it cannot truly said to be "built"
Then if completed infinities (or irrational numbers) are finished houses, they do not yet exist, constructively speaking. | |
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| Philosophies of math...Includes essay...seriously
Posted: 11/6/2009 11:33:32 AM | RE Msg: 34 by JustDukky:
we are left with the fundamental problem that infiinity has as much validity as irrational numbers. Yet, everyone seems to accept that irrational numbers exist. Who is "everyone"? An intuitionist would not assume the a priori "existence" of a number that cannot be exactly specified or proved. For instance, Brouwer showed that an exact value of "Pi" cannot be constructed, so in what sense does "Pi" exist for the constructivist/intuitionist? Classical mathematics "proves" the existence of (for instance) the square root of two, but intuitionist mathematics does not assume it exists because it "must". The square root of two must be constructively proved without using the law of the excluded middle. (i.e. it is not sufficient to say it can't "not exist" therefore it must exist)/ As Kronecker once said to Lindemann: "Of what use is your beautiful investigation regarding Pi? Why study such problems, since irrational numbers are non-existent?" So I can think of at least one mathematician you'd have to exclude from the set called "everybody."
However, each has their limits. There are things we can prove formally, that we cannot prove constructively, yet intuitively see makes sense, such that the ratio between the circumference of a circle and its radius appear to be constant for all circles, and therefore exists as an actual number, but also appear to be unable to be expressed in rational terms. There are things that make sense in intuitionism that do not make sense in formalism. I imagine that there are things that can be proved using constructionism that cannot be proved formally, such as the algorithm for proving that every vector space has a basis. Each has much to teach us.
However, what distinguishes mathematics from other subjects, is not its philosophical underpinnings, but its requirement for everything in mathematics to be proved unquestionably. If there is any possibility, no matter how far removed, that an alternative to a given theorem might exist, then the theorem cannot said to be true in mathematics. It is that high requirement of almost infinite rigour, that makes mathematics so reliable. It equally means that mathematical theorems are so abstract, that they can only be an approximation to real life. But the approximation can be so close to the axioms of a theorem, that the results can be extremely close to the reality, and we are even able to calculate with great accuracy exactly how far we can expect reality to differ from the results of our theorem, based on how far our axioms might differ from our measurements of those axioms in real life.
We CAN be extremely pedantic about all this. We CAN demand that mathematics is formalistic, or constructionist, or intuitionist, and argue ad infinitum. But then, we're more interested in the argument than getting to what is really going on, and what works best. OR, we can simply contend that right now, we just cannot exactly say what is the philosophy underpinning mathematics, and that we don't really need to, as long as we stick to what mathematics really gives us, incredibly reliable methods, by being extremely rigorous about our proof of those arguments.
Once you have the bricks and the plans, it is in the process of being built. However, until it is finished, it cannot truly said to be "built" Then if completed infinities (or irrational numbers) are finished houses, they do not yet exist, constructively speaking.
Once you have the bricks and the plans, it is in the process of being built. However, until it is finished, it cannot truly said to be "built", even if the house is being lived in right now. Can one live in a house that doesn't exist? We have people living in seriously incompleted housing right now. We might be inclined to say they're not liveable, as ther certainly are far from it. But couples starting such renovations are living in such abodes. They aren't finished houses. Some people would day they are living in a shack and not a house, and refuse to ever visit them. Others would say that they're living in a house that needs work, and still comes round for dinner. The real question is what are we going to do about it? If they live in a hovel, or an unfinished house, will you help them finish it, but still visit them and let them enjoy your company, or will you lord it over them for not living in a real house?
I'm all for discussions about mathematics. But not for meaningless arguments of "I'm right, and you're wrong", certainly not in mathematics. I've seen what happens all too often when that attitude is taken in other subjects. You just get long-winded arguments, end up p*ssing off everyone you argue with, and nothing really gets resolved. Mathematics deals with the abstract. The only thing it does give us is clear methods for use in approximating reality. Take that away, and it ceases to be useful to us at all.
But thanks for the discussion. It's given me much more food for thought. | |
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| Philosophies of math...Includes essay...seriously
Posted: 11/6/2009 12:14:20 PM | issues like the existence of infinity, do generally seem to be intuitively true, based on constructivist methods
?????????... That makes no sense.
Infinities can't be constructed and therefore cannot be considered "true" in constructive terms. Since they cannot be built, they cannot exist. That is the whole essence of intuitionist philosophy. (I KNEW using the word "intuitionist" would cause problems!...Looks like I was right.) Intuitionists DO NOT believe things because they are intuitively true (at least not in the sense of what we nomally call "intuition"). They believe mathematics is a product of the mind and therefore of intuition. They DO NOT believe things to be "intuitively true". The mathematics itself is intuited, but mathematical objects must be PROVED in order to "exist" at all. It is NOT SUFFICIENT to be "intuitively true". unless by intuition you mean the formal reasoning of thought.
although it seems to be true most of the time, seems also to be untrue in some rare cases, like the "set of all sets", and so is also just as useful
This doesn't make sense either. Whether or not something is "true" (in this case the law of the excluded middle) is entirely a matter of the philosophical perspective, it is never considered "true" in some cases and "untrue" in others. Whether or not it is valid is a matter of one's philosophy of mathematics, not of mathematics itself.
Re: the "set of all sets"; didn't you mean "useless"? The "set of all sets" was the mess that forced the creation of ZF set theory, an ugly kludge that kept the bricks & mortar of mathematics from crashing down around us. That's what happens when unproved assumptions are made and why the intuitionists don't allow them simply because they seem "intuitively true."
what distinguishes mathematics from other subjects, is not its philosophical underpinnings, but its requirement for everything in mathematics to be proved unquestionably
That is the intuitionist argument. There should be NO ASSUMPTIONS beyond the axioms themselves and the axioms ought to even have a firm logical basis in that they must be shown not to conflict with the other axioms...ever.
we just cannot exactly say what is the philosophy underpinning mathematics
That's like asking which religion is the underpinning of theology. Lotsa luck on that one.
The real question is what are we going to do about it? If they live in a hovel, or an unfinished house, will you help them finish it
An infinity is a house whose construction can never be finished. The best you can do is let people live in whatever they can build, so long as they are aware that the roof doesn't exist and can never be put on.
I'm all for discussions about mathematics. But not for meaningless arguments of "I'm right, and you're wrong", certainly not in mathematics.
So who's arguing?...I'm just explaining the different mathematical philosophies. | |
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| Philosophies of math...Includes essay...seriously
Posted: 11/6/2009 1:46:17 PM |
Infinities can't be constructed and therefore cannot be considered "true" in constructive terms.
Lets move on a little.
We as humans are very limited creatures. When we measure or construct something we have to do it in time. There is a major problem here. The problem is that we are stuck with the now and the ability to see the past. We are blind to the future and can only infer it through observation of the past.
It is my opinion that any mathematician must see reality as a system that is predictable by the logical observation of the past. If a ball drops then this ball must drop. We could not draw such a conclusion without past observation.
So my point is this. Reality must be a system. To understand the system we must be aware of all past fundamental observations. Lets call these observations axioms. The problem is that we are in the system and as such it is impossible to be aware of all the system. We cannot see the entirety of time. We cannot see infinity, we can't even see "pi" fully. All we can see is finiteness. So the mare fact that infinity is applicable within the system is an indicator of its existence. BUT, along with Godel's incompleteness theorem, we do not have full awareness of infinity. Just certain attributes of it.
It follows that the best way to know the system is to see all the system. We humans cannot do that, because our senses are limited to the five senses. Just like we can't see 5 spatial dimensions. So while I do believe that the answer exits before we find it, I also believe it is impossible to precisely know the answer because of the nature of our existence.
Basically the platonist and the constructivist are right. The pure answer exists (platonist) and it is possible to know, but only through a complete perception of all of reality. However, the pure answer is impossible to know for us humans since we do not see the entirety of reality. Therefore mathematical prediction of the future or any model explaining the system of reality is a product of intuition and not necessarily true for all reality.
Is this a logical conclusion to the unification of two philosophical ideas within mathematics? | |
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| Philosophies of math...Includes essay...seriously
Posted: 11/6/2009 3:38:48 PM | RE Msg: 36 by JustDukky:
issues like the existence of infinity, do generally seem to be intuitively true, based on constructivist methods ?????????... That makes no sense. Infinities can't be constructed and therefore cannot be considered "true" in constructive terms. Since they cannot be built, they cannot exist.
That is the whole essence of intuitionist philosophy. (I KNEW using the word "intuitionist" would cause problems!...Looks like I was right.) Intuitionists DO NOT believe things because they are intuitively true (at least not in the sense of what we nomally call "intuition"). They believe mathematics is a product of the mind and therefore of intuition. They DO NOT believe things to be "intuitively true". The mathematics itself is intuited, but mathematical objects must be PROVED in order to "exist" at all. It is NOT SUFFICIENT to be "intuitively true". unless by intuition you mean the formal reasoning of thought. Intuition is a product of the mind, without needing to be explained such as in a formal proof. Thus, what is known as "intuitively true", is that which one's mind tells one must be true, without the first existence of a formal proof, but that with such intuition, such as that Fermat's Last Theorem is true, one has the confidence that a proof exists, and one simply sets out to find it. It is the process of finding your OWN solutions to problems, and caring not if others have solutions that you could just accept instead.
I've always been told that I was intuitive in mathematics. I was told many times it was not the norm. I also found that most non-intuitives simply didn't grasp that if there was another way of doing it that wasn't in the book, that it could be equally right, and always demanded that if it wasn't the formal proof they were taught, that it was wrong. Equally, non-intuitives seem to have no idea how someone can come up with a problem to a solution that isn't written down, and seem to think that such people are "magical". What you are describing is a belief that many non-intuitives seem to have, that we cannot trust our own intuition, that we must always stick to constructions, and not think outside the box, and that those before us who come up with mathematical theorems must develop them from some hidden recess of their minds, that they label "intuition". If anything, I would call the philosophy that you have described, "non-intuitionism", because it fits perfectly with what I have heard expressed from many people who are open that they simply couldn't come up with an original solution to a maths problem if they tried for 100 years.
although it seems to be true most of the time, seems also to be untrue in some rare cases, like the "set of all sets", and so is also just as useful This doesn't make sense either. Whether or not something is "true" (in this case the law of the excluded middle) is entirely a matter of the philosophical perspective, it is never considered "true" in some cases and "untrue" in others. Whether or not it is valid is a matter of one's philosophy of mathematics, not of mathematics itself.
Re: the "set of all sets"; didn't you mean "useless"? The "set of all sets" was the mess that forced the creation of ZF set theory, an ugly kludge that kept the bricks & mortar of mathematics from crashing down around us. No. Frege developed axiomatic set theory in 1879, based on the fundamentals that we took for granted in the last 500 years of deterministic empirical philosophy. Cantor published his theorem in 1891. Russell studied Cantor, and by 1901 published that Cantor's work showed a huge flaw in axiomatic set theory. The "set of all sets" was just a way of categorising ALL infinities. But it still contained the flaw, and the only resolution was that the Law of the Excluded Middle could not apply to it, and Cantor's Theorem must apply to it, and yet it must not be true to it, at the same time. But that meant that there was a clear concept of something that didn't fit into propositional logic and set theory, and really broke all the rules of mathematics as it was understood, by anyone. The Formalist solution expressed in terms of Zermelo's axioms was an attempt to describe a system of rules in which Russell's antimony would not have to apply. The Constructionist solution was an attempt to describe a set of methods of proof in which Russell's antimony would not apply. But fundamentally, the problem didn't come from Russell at all. It didn't come from Cantor's theorem. It came from the PROOF of Cantor's theorem, from the logic he used, which was that any time you could conceivably construct any set based on a negative self-referential function, then you automatically were led to the same conclusion, irrespective of the field. Cantor's theorem doesn't just apply in set theory. Rather, it applies all over mathematics. The argument can even be used beyond mathematics in real life. What is more, because Cantor used pre-existing notions that everyone agreed were perfectly valid ideas, "a is a member of S", "f(x)", etc, there really is nothing in Cantor's theorem that you cannot find in any field of mathematics. Even the approach was universal, because it was based on a very simple but ingenious construction.
You can eliminate all sorts of ideas. But fundamentally, Cantor's theorem speaks to the heart of mathematics, and the heart of logic itself. It can be summed up by "I am lying. How can you tell?" Answer: You can't. You need some piece of evidence to TELL you that I'm lying. But how do you know that that evidence isn't lying, isn't false, and I'm the one speaking the truth? Answer: You can't. All you can do is say that if you have 10 people telling you I'm lying, and only 1 person saying I'm telling the truth, that you'd bet that there is more of a chance of 10 people being right than 1 person being right. It's betting, and that's not very mathematical at all.
That's what happens when unproved assumptions are made and why the intuitionists don't allow them simply because they seem "intuitively true." Formalists don't either. The only difference is that formalists consider that ALL things in mathematics come from a minimal set of axioms, and they cannot be proved, no matter what. Thus, to a formalist, mathematics doesn't say "Y is true", but "if X is true, then Y is true", and if X is true in the real world, then so is Y. Intuitionism appears to assume that you cannot rely on the Law of the Excluded Middle, and what can be proved without it, is obviously an advantage. But it's still just another model, with one less axiom. Some things will be provable in formalism that cannot be proved in constructionism, because it lacks the Law of the Excluded Middle. That just means that where the Law of the Excluded Middle seems to be true IRL, that we can rely on Formalism as well as Constructionism, and gain the reliable use of those extra theorems.
what distinguishes mathematics from other subjects, is not its philosophical underpinnings, but its requirement for everything in mathematics to be proved unquestionably That is the intuitionist argument. There should be NO ASSUMPTIONS beyond the axioms themselves and the axioms ought to even have a firm logical basis in that they must be shown not to conflict with the other axioms...ever.
we just cannot exactly say what is the philosophy underpinning mathematics That's like asking which religion is the underpinning of theology. Lotsa luck on that one.
The real question is what are we going to do about it? If they live in a hovel, or an unfinished house, will you help them finish it An infinity is a house whose construction can never be finished. The best you can do is let people live in whatever they can build, so long as they are aware that the roof doesn't exist and can never be put on.
I'm all for discussions about mathematics. But not for meaningless arguments of "I'm right, and you're wrong", certainly not in mathematics. So who's arguing?...I'm just explaining the different mathematical philosophies. | |
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| Philosophies of math...Includes essay...seriously
Posted: 11/7/2009 1:52:24 AM |
Infinities can't be constructed and therefore cannot be considered "true" in constructive terms. Since they cannot be built, they cannot exist. But they are built.
An algorithm is a construction, yes, but it in the case of infinities it is not a COMPLETED construction. Maybe the best way to put it is in terms of a Turing Machine. If it is set to the task of computing Pi exactly, it will never halt (because there is no last decimal place, nor a rule which defines all following numbers in the decimal in a finite amount of time. If it never halts, the construction is never completed. If it is never completed, it cannot be said to have any particular value beyond an approximation. If exactly Pi cannot be constructed, how can it be said to exist (except in a world of forms, independent of mind)? The intuitionist would say "it only exists if I can construct it. Since I can't, it doesn't exist." The formalist would say (essentially), "Who cares? mathematics is only a game I play in which I make the rules, so if I say it "exists", it "exists" because I have said it does, unless you can prove that the rules of the game prohibit its presence on the "board" and therefore can't be allowed to "exist." It should be noted that the formalist is "anti-Platonic" in that he doesn't believe anything in mathematics really exists; mathematics is just a game of symbols and rules that he makes up and allows (but most formalists wouldn't want to throw away something they find useful, even if its use might yield inconsistency - There are exceptions, as it may be said that constructivism can be considered "strong formalism" - Read the work of Wittgenstein - fascinating!)
This is not a matter of the "right" perspective of mathematics. It is a matter only of philosophical perspective. (Yet) another analogy might be in moral theory. Which is "right", Consequentialism, or Deontology? What defines the morality of an action, its consequence, or its intent? Obviously you'll get differing answers based on the philosophies of the people you ask.
mathematics is more than just about personal philosophies. It's about finding solutions irrespective of your personal philosophy
The personal philosophies are nevertheless relevent to mathematics itself. Joe mathematician may only be a "mechanic" finding solutions by using the tools, but the mathematical philosopher is concerned with mathematics itself and that has a great bearing on the tools Joe uses, his confidence in the solutions he finds and what he thinks he's doing. | |
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| Philosophies of math...Includes essay...seriously
Posted: 11/7/2009 4:49:20 PM | RE Msg: 39 by JustDukky:
An algorithm is a construction, yes, but it in the case of infinities it is not a COMPLETED construction. Maybe the best way to put it is in terms of a Turing Machine. If it is set to the task of computing Pi exactly, it will never halt (because there is no last decimal place, nor a rule which defines all following numbers in the decimal in a finite amount of time. If it never halts, the construction is never completed. If it is never completed, it cannot be said to have any particular value beyond an approximation. I quite agree. That's exactly why my maths teacher taught us to never rely on calculators. But in formalism, the multi-digit system of expression is only one form of expression. It carries no more or less weight than -i*ln(-1). Otherwise, formalism couldn't deduce anything that didn't have a finite number of digits in base 10, like 1/3.
The intuitionist would say "it only exists if I can construct it. Since I can't, it doesn't exist." The formalist would say (essentially), "Who cares? mathematics is only a game I play in which I make the rules, so if I say it "exists", it "exists" because I have said it does, unless you can prove that the rules of the game prohibit its presence on the "board" and therefore can't be allowed to "exist." It should be noted that the formalist is "anti-Platonic" in that he doesn't believe anything in mathematics really exists; mathematics is just a game of symbols and rules that he makes up and allows (but most formalists wouldn't want to throw away something they find useful, even if its use might yield inconsistency - There are exceptions, as it may be said that constructivism can be considered "strong formalism" - Read the work of Wittgenstein - fascinating!) The formalist would simply point out that if you set a traditional view of a computational Turing machine to to the task of computing 1/3, then it too would never halt. But then, we don't really need to be that restrictive over the type of results that any Turing machine would produce. We can simply say that 1.2 is a functional expression of 1+2/10, and is no more valid than 1/3. Moreover, since we can say that 1/3=1.333333 recurring, we can point out that Pi has the same number of decimal places as 1/3, and yet 1/3 is just as valid as 1.2
However, I would agree that formalists just play games, in that they can be said to be using rules of string manipulation. However, that's pretty much what we do with computers, because all data in a computer is a string, that is manipulated using a series of rules that are applied again and again in varying patterns. We can also note that the brain works in a pattern-matching way, that manipulates data in very similar ways to string manipulation rules. In that way, the formal system does resemble exactly what the brain does, and since formal proofs are developed by humans, is a product of the mind, and a close approximation to the mind's method of problem-solving.
But the brain also uses constructive methods. So I would not want to claim that formalism is "better" than constructivism, just that each are "different", and each are useful, each valid, no one better than the other.
This is not a matter of the "right" perspective of mathematics. It is a matter only of philosophical perspective. (Yet) another analogy might be in moral theory. Which is "right", Consequentialism, or Deontology? What defines the morality of an action, its consequence, or its intent? Obviously you'll get differing answers based on the philosophies of the people you ask. You're right. It depends on who you ask. However, I think you'll find some consequentialism in deontology, like that murder is OK if it's in self-defence. I expect you'll also find some deontology in consequentialism. Ultimately, all perspectives usually have a bit of each perspective. It's more a question of which is your priority? Do you put consequentialism before deontology, or deontology before consequentialism?
I know that formal mathematics sometimes uses constructivist methods. I would be very surprised if constructivism doesn't use anything in formalism. So again, I think it's probably a question of if you put constructivism before formalism, or formalism before constructivism?
The personal philosophies are nevertheless relevent to mathematics itself. Joe mathematician may only be a "mechanic" finding solutions by using the tools, but the mathematical philosopher is concerned with mathematics itself and that has a great bearing on the tools Joe uses, his confidence in the solutions he finds and what he thinks he's doing. I'd rather not be so adamant that one philosophy should take precedence over another. I used to think so, and I realised it cost me greatly. So these days, I'm more apt to say "whatever works". If constructivism can teach us something about mathematics that we cannot learn via formalism, or that we can, but it is shown much clearer in constructivism, then use that. Equally, I would say the reverse, that if formalism can teach us better than constructivism about something, then use that. Maybe that makes me on the fence. Maybe that makes me sound childish. I don't really care any more. I think: if it works best, use it, because that's more efficient, and more productive, to everyone. | |
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| Philosophies of math...Includes essay...seriously
Posted: 11/7/2009 7:19:07 PM | if you set a traditional view of a computational Turing machine to to the task of computing 1/3, then it too would never halt
It would halt almost immediately. I suggest you do some reading on Turing machines and the theory behind them.
they can be said to be using rules of string manipulation. However, that's pretty much what we do with computers
Yes it is.
I would not want to claim that formalism is "better" than constructivism, just that each are "different"
Actually, constructivism could be considered a subset of formalism that we might call "strong formalism", as I pointed out in a prior post.
I think you'll find some consequentialism in deontology, like that murder is OK if it's in self-defence. I expect you'll also find some deontology in consequentialism. Ultimately, all perspectives usually have a bit of each perspective
That people may incorporate elements of both philosophies, or have a personally inconsistent philosophy and "not know (or care) what they are" is immaterial; the philosophies are fundamentally different. Murder is a bad example to use because it is always wrong to murder; I think you really meant homicide, which is sometimes justifiable (like in self defense). In all fairness though, murder could be seen as morally right by a consequentialist. If an assassin murdered Hitler and the consequence was averting a war that would have killed millions, the consequentialist wouldn't give two hoots about the assassin's motive, he'd probably get a medal. A deontologist would look at the motive (say money) and determine the action immoral whatever the consequence. The assassin would probably hang.
I know that formal mathematics sometimes uses constructivist methods. I would be very surprised if constructivism doesn't use anything in formalism
The first sentence is true. The second sentence is silly. All of the constructivists methods are formal. In fact one could say that constructivists are more formal than the formalists in that they'd never wear a sport jacket to a "tails" event. The constructivist has built his wardrobe in such a way that there is no possibility of putting on something that isn't coordinated with the rest of the outfit.
I'd rather not be so adamant that one philosophy should take precedence over another.
Who's doing that? Did I say I thought Gödel was stupid for being a Platonist? Did I say that Hilbert was only in the game for laughs? Did I say Kronecker was a crank? I'm only trying to explore & illustrate the different philosophical perspectives held by mathematicians. I thought this would be a good forum for it because most people are entirely unaware of the extreme differences in philosophy in what looks like a pretty exact science where everything appears well established & harmonious.
I just thought of another way to describe the philosophical differences:
The Platonist: God exists and we can discover some things about him.
The Formalist: There is no God, but we can create one on paper with the right inferences.
The constructivist: Inference my ass! God isn't constructable and therefore can't exist, even on paper. | |
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| Philosophies of math...Includes essay...seriously
Posted: 11/8/2009 8:08:12 AM |
The pure answer exists (platonist) and it is possible to know, but only through a complete perception of all of reality. However, the pure answer is impossible to know for us humans since we do not see the entirety of reality. Therefore mathematical prediction of the future or any model explaining the system of reality is a product of intuition and not necessarily true for all reality.
Is this a logical conclusion to the unification of two philosophical ideas within mathematics?
The problem is threefold. In the first place, we can't know whether or not reality matches our perceptions of it. The second problem stems from using our perceptions of reality as the basis for descriptive language. The third problem is usng definitions that may not accurately describe reality as self-evident truths (axioms). Let's explore this with a simple concept used as a definition:
A bunch of rocks on the ground is observed and the caveman mathematician says "I think I'll define the "set"; "A set is a collection of objects." Of course he now must define in precise terms what a collection is and what an object is, but we'll skip that for the moment.
Now let's say we have a really smart, mathematically inclined slug climbing over those rocks. He's blind and can only sense what he actually crawls over. He notices a difference between the height and texture of the ground vs the rocks and perceives the difference in mathematical terms. He might think rock differs from earth in different ways than the man would. He might think "a set is a consecutive number of lumps of differently textured substrate". We note two differences immediately from the man's definition. One: The man perceives discrete "objects", separate from the earth they sit upon. Two: The man doesn't assume the rocks to be "ordered", but the slug does. So already, we see two significantly different definitions arising from the same reality as a result of different perceptions of it. Suffice it to say that a slug's perception of reality differs markedly from our own. Do both definitions correspond to reality? Yes they do, but in different ways. Either one could be used as a definition for an axiomatic set theory, but I suspect the axioms would be completely different in each case, which would result in different set theories. Q: Which one is modeled on reality? A: Neither; they are both modeled on perceptions of reality.
Q: Does either set theory correspond with reality?
A: If we don't know objectively what reality is, how can we know? The best we can do is try to build a system of mathematics that is consistent and doesn't contradict itself. That system probably has some correspondence to reality, but being based on erroneous(?) perceptions of it, probably has vast areas which have no correspondence whatsoever.
Even if one of the systems of reasoning DID correspond perfectly with physical reality (and the other did not), they would BOTH be mathematics, so this is why I say mathematics is much "richer" than "reality" in that the mathematics of "reality" is only a subset of all mathematics. Here we have a bit of a quandary for the Platonist; does the world of forms contain all of mathematics, or only a "perfect" reality? If the Platonist intuits something from the world of forms, how can he be assured it corresponds with "reality"? The world of forms then, must be a complete collection of all potential realities and the Platonist must "choose wisely" when he intuits it if he is to build a system that is not at odds with itself.
Here we can "unite" Platonism with formalism. Since the world of forms contains every possible mathematical object, there is nothing the formalist can create that isn't consistent with the world of forms. Since the world of forms "exists" only in terms of potential realities, it doesn't really exist, so the formalist is right in that sense. This also creates an interesting question...How can a formalist accept Gödel's incompleteness theorems without being a Platonist? This would naturally create a "schism" in formalism (it already existed, but Gödel's theorems fit better to illustrate it). On the one hand you would have some formalists (the majority) saying that some mathematical objects create problems of consistency with others, but it is all mathematics and many of the objects are too useful to discard for the sake of consistency. They would agree that Gödel's theorems can be worked with "as though" they were "objectively true", since all they are doing is pushing symbols around on paper in a way that really needs no correspondence to reality or "truth". Truth in mathematics can now be seen as a sort of "pseudo-truth" in a game of "pseudo-reality" called mathematics.
Some formalists felt that the most important thing in mathematics was consistency. Anything that might produce inconsistency ought to be discarded, or the game itself would become worthless and meaningless. These are the constructivists/intuitionists.
I feel at this point there has been some confusion regarding "intuition" introduced by my use of the term "intuitionist". Both the Platonists and constructivists use intuition, but it has a different definition, or meaning to each philosophy. A Platonist who "intuits" mathematics believes his mind has PERCEIVED something that already exists. The constructivist/intuitionist denies the "pre-existence" of mathematics and when he "intuits" a mathematical object, he means the process of his mind has CREATED it. The formalist would agree with the constructivist view, but would add that mathematics can be created by inference "as though" it pre-existed. (Sort of like "Hey, it's just a game of symbols and I write the rules, so why can't I just assume things into existence in my imaginary little world?")
I guess another difference that might distinguish the constructivist from the formalist is that I suspect the constructivist believes he has really built something real, where weaker formalists, in allowing contradictory (or at least problematic) elements to exist in the system, can't claim his assumptions have built anything meaningful, so he claims he's not building anything meaningful and pretends math is all make-believe. (of course if he really believed it was make-believe, he probably wouldn't waste his time with it, so I suspect this "just a meaningless game of symbols & rules" crap is just an apology to keep his a$$ covered when pressed about the inconsistencyof his pseudo-platonic "faith" in a meaningless game.
Did I say somewhere that most formalist mathematicians are "closet platonists"? I read somewhere that a mathematician is a platonist when he's working and a formalist when asked about his work. I think that's true. | |
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| Philosophies of math...Includes essay...seriously
Posted: 11/8/2009 8:50:40 AM |
Did I say somewhere that most formalist mathematicians are "closet platonists"? I read somewhere that a mathematician is a platonist when he's working and a formalist when asked about his work. I think that's true.
I agree
Paul Dirac:
"The very idea of God is a product of the human imagination...
"...the honest assertion that God is a mere product of the human imagination is branded as the worst of all mortal sins."
And that was from the great Paul Dirac. | |
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| Philosophies of math...Includes essay...seriously
Posted: 11/8/2009 9:19:00 AM | | The guy I feel sorriest for is Gödel (one of my heroes). I think he spent the latter part of his life trying to prove the existence of the unprovable and it was likely the torment and obsession of that effort that drove him nuts. Intuitionists/constructivists wouldn't "waste their time & effort" trying to construct the unconstructable and probably manage to keep a better grip on their sanity as a consequence. | |
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| Philosophies of math...Includes essay...seriously
Posted: 11/8/2009 10:10:58 AM | RE Msg: 41 by JustDukky:
Actually, constructivism could be considered a subset of formalism that we might call "strong formalism", as I pointed out in a prior post. That's very different from a different philosophy, as then constructivists would agree with everything that formalists agree, and would be formalists themselves, only that they have some additional requirements that make them more rigorous than formalists. Also, that would equally mean that everything that constructivists do, is automatically true for formalists. However, results of constructions are things that are not necessarily true in a formal setting, and need to proved formally.
What you wrote here, might indeed be true. But really, only a group of formalists could be relied upon to testify that they consider constructivism as a stronger form of formalism. If you have a site of formalists, that do indeed say exacly this, that would be proof enough for me.
I know that formal mathematics sometimes uses constructivist methods. I would be very surprised if constructivism doesn't use anything in formalism The first sentence is true. The second sentence is silly. All of the constructivists methods are formal. In fact one could say that constructivists are more formal than the formalists in that they'd never wear a sport jacket to a "tails" event. The constructivist has built his wardrobe in such a way that there is no possibility of putting on something that isn't coordinated with the rest of the outfit.
I just thought of another way to describe the philosophical differences:
The Platonist: God exists and we can discover some things about him.
The Formalist: There is no God, but we can create one on paper with the right inferences.
The constructivist: Inference my ass! God isn't constructable and therefore can't exist, even on paper. Everything I know about maths says that ALL of those would be wrong, because you cannot say anything does or doesn't exist in mathematics, unless you have a solid proof of it. I really don't know how to make sense of what you are saying.
Neither would I consider anything that is non-constructable as being the requirement for existence, because AFAIK, the integers are not constructable, and nor is anything else in mathematics. I realise that we are probably going to go at loggerheads over this. So a link to the exact definitions of constructivism and a simple proof that the integers are constructable would solve this point.
I'm really not interested in debating about philosophy ad nauseum. That's why I got into maths and not philosophy. I found that philosophy involved arguments that just continued and continued, and everyone who ever studied it, told me that you went into it with questions, and just came out with more. With maths, you went in with questions and you came out with clear answers. So I'd rather read a properly rigorous mathematical proof of what you are claiming. At least with mathematics, SOMEONE will probably have written such proofs.
RE Msg: 44 by JustDukky:
The guy I feel sorriest for is Gödel (one of my heroes). I think he spent the latter part of his life trying to prove the existence of the unprovable and it was likely the torment and obsession of that effort that drove him nuts. Intuitionists/constructivists wouldn't "waste their time & effort" trying to construct the unconstructable and probably manage to keep a better grip on their sanity as a consequence. Why? Gödel is one of my heroes. I've spent several years trying to grasp the underpinnings of his theorems and what consequences they have for mathematics, and for the world. The only problem with Gödel is that he was way ahead of his time, and the world is not even close to catching up. He's on a par with Einstein. | |
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| Philosophies of math...Includes essay...seriously
Posted: 11/8/2009 12:00:05 PM | That's very different from a different philosophy, as then constructivists would agree with everything that formalists agree, and would be formalists themselves, only that they have some additional requirements that make them more rigorous than formalists. Also, that would equally mean that everything that constructivists do, is automatically true for formalists. However, results of constructions are things that are not necessarily true in a formal setting, and need to proved formally.
Scorp...What are you not getting? I'm running out of ways to explain the similarities & differences between the philosophies in such a way that you understand. Maybe the best way to put it is to say that constructivists/intuitionsts ARE formalists who think that most formalists are too informal about their reasoning by letting things into the system that are only implied to exist by what they'd call platonic acts of faith.
That's very different from a different philosophy, as then constructivists would agree with everything that formalists agree, and would be formalists themselves, only that they have some additional requirements that make them more rigorous than formalists. Also, that would equally mean that everything that constructivists do, is automatically true for formalists. However, results of constructions are things that are not necessarily true in a formal setting, and need to proved formally.
I'll do that when you can show me an article from the journal of deontological consequentialism.
a sufficient perfect proof of this would suffice, such as a link to a website listing all the axioms of formalism, and the axioms of constructivism, and how constructivism must be a subset of formalism.
I guess denial of accepted formalisms like the law of the excluded middle, the law of trichotomy, and irrational numbers doesn't quite "cut it" for you then, eh?
you cannot say anything does or doesn't exist in mathematics, unless you have a solid proof of it
I guess you pity those poor platonist & formalist mathematicians who start off proofs with unproved assumptions like "There exists...". They should know better (after all, the constructivists know better than that, right?)
I really don't know how to make sense of what you are saying.
I know, and it's getting frustrating.
the integers are not constructable
Yes they are, only the constructivists will not concede an infinite set of them, only an arbitrarily large finite set (i.e. "n", but not "aleph null"). You have to bear in mind that to the constructivist "not finite" DOES NOT IMPLY "infinite" because they reject the law of the excluded middle.
the integers are not constructable
What math did YOU study???
I'd rather read a properly rigorous mathematical proof of what you are claiming.
What am I claiming beyond the existence of different schools of mathematical thought? Are you asking me for an "existence proof" of them? If you are, what if I'm a constuctivist and can't give you a valid proof of my existence?
He's on a par with Einstein.
Gödel was (arguably the greatest logician who ever lived and was) MUCH smarter than Einsein. I revere Einstein for his great humanity and compassion, not so much for his brain; many of his contemporaries were smarter. | |
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| Philosophies of math...Includes essay...seriously
Posted: 11/8/2009 12:13:09 PM |
I just thought of another way to describe the philosophical differences:
The Platonist: God exists and we can discover some things about him.
The Formalist: There is no God, but we can create one on paper with the right inferences.
The constructivist: Inference my ass! God isn't constructable and therefore can't exist, even on paper.
Given the problem of proving or disproving the existence of God through mathematics.
The platonist: There is a solution to this conundrum and along the way to the solution I will learn more about other things.
The formalist: There is a solution but I need to use the prior rules of math to solve it. If it is unsolvable than it means I must resolve this issue by finding the theorems or axioms that do allow for a solution. As far as I know, the solution does not refer to reality.
The constructivist: God eh? Poppy**** Mathematics is consistent to observation. We do not consistently observe God and everything works purely on the mechanics of the natural universe. Therefore God is an unobserved and does not relates in anyway to consistent observation. God cannot exist and mathematics cannot be used to infer the existence of the unobserved.
The philosophies are important to any mathematician seeking solutions to problems. I doubt Godel would have the incompleteness theorem if he didn't embrace the philophosy o platonist.
I've spent several years trying to grasp the underpinnings of his theorems and what consequences they have for mathematics, and for the world. The only problem with Gödel is that he was way ahead of his time, and the world is not even close to catching up. He's on a par with Einstein.
It seems you are a platonist. I imagine a constructivist would be of the opinion that you are wasting you're time. Also, the statement "...trying to grasp the underpinnings of his theorems and what consequences they have for mathematics, and for the world." Is philosophical in its nature.
You can't relate the abstract to reality without dabbling in philosophy. Since the abstract, as far as we know, may not relate to reality. | |
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| Philosophies of math...Includes essay...seriously
Posted: 11/8/2009 1:00:17 PM | RE Msg: 46 by JustDukky:
Scorp...What are you not getting? I'm running out of ways to explain the similarities & differences between the philosophies in such a way that you understand. Maybe the best way to put it is to say that constructivists/intuitionsts ARE formalists who think that most formalists are too informal about their reasoning by letting things into the system that are only implied to exist by what they'd call platonic acts of faith. That's words, and more importantly, English words, and I know that English is one of the most vauge and ambiguous languages out there, if not THE most vauge and ambiguous language. I find what you are saying is nowhere near rigorous enough for me to apply, except in the most vague and ambiguous way. That might work when dealing with people in general, and I really am happy to let you say it. But I cannot work with that in the rigour of mathematics, at least the way I was taught to use it. It's unworkable for me in mathematics. But if you put the same into a proper set of mathematical symbols, such as formalists = { p e People : formalist(p) = True } and constructivist = { p e People : constructivist(p) = True }, defining People = { the set of all people } and defining formalist(p):People -> {True, False} and defining constructivist(p):People -> {True, False}, then I might be able to work with that with mathematics.
Sorry. I guess I'm just nowhere near as smart as you. Help a stupido out?
I'll do that when you can show me an article from the journal of deontological consequentialism. I didn't bring it up. I really only discussed it in terms of what you said. I could no more bring up an article on deontological consequentialism, than I could prove the value of constructionism to anyone. Maybe I should just accept that constructionivism is just something that I have to ignore?
I guess denial of accepted formalisms like the law of the excluded middle, the law of trichotomy, and irrational numbers doesn't quite "cut it" for you then, eh? No. "like" is not really very clear, is it? I don't really know what's IN, and what's OUT, with constructivists. I also don't know how they understand arithmetic without trichotomy and irrational numbers. I also don't know if the removal of trichotomy and irrational numbers is due to the removal of the law of the excluded middle, or down to something else.
I guess you pity those poor platonist & formalist mathematicians who start off proofs with unproved assumptions like "There exists...". They should know better (after all, the constructivists know better than that, right?) I hope not. I've not proved constructivism exists either. I would have to tell myself off for even being willing to discuss something that I have never shown exists.
I really don't know how to make sense of what you are saying. I know, and it's getting frustrating.
the integers are not constructable Yes they are, only the constructivists will not concede an infinite set of them, only an arbitrarily large finite set (i.e. "n", but not "aleph null"). You have to bear in mind that to the constructivist "not finite" DOES NOT IMPLY "infinite" because they reject the law of the excluded middle.
the integers are not constructable What math did YOU study???
Gödel was (arguably the greatest logician who ever lived and was) MUCH smarter than Einsein. I cannot say that for sure. But I do agree that he was a very smart dude.
I revere Einstein for his great humanity and compassion, not so much for his brain; many of his contemporaries were smarter. I haven't read much by Einstein that gives me the impression that he thought too deeply about either. I haven't read that much that implies that he was all that compassionate either. I have read that he was really great at Physics, and that he was realistic enough to realise that his actions had consequences. But I really cannot confirm much more than that.
RE Msg: 47 by exogenist:
I've spent several years trying to grasp the underpinnings of his theorems and what consequences they have for mathematics, and for the world. The only problem with Gödel is that he was way ahead of his time, and the world is not even close to catching up. He's on a par with Einstein. It seems you are a platonist.
I imagine a constructivist would be of the opinion that you are wasting you're time. Also, the statement "...trying to grasp the underpinnings of his theorems and what consequences they have for mathematics, and for the world." Is philosophical in its nature. It does sound so. But really, it's rather like saying that you've proved that a particular type of economic system (capitalism, communism, whatever) will always destroy the economy that it runs on. It doesn't just tell you something about the economic system. It tells you everything that uses the same basic principles, as it becomes a mathematical model for many subjects that have the same axioms, and in which the same results can be expected to occur.
Mind you, if you try to pin me down to some sort of philosophy, you're bound to fail. I work the other way around. I start off with a few things, and build on that. If that results in a conclusion that contradicts one of the initial things I started with, I check my logic, and if I cannot find anything wrong with it, I make my initial concept more general. So, my philosophy of mathematics has expanded year on year, and carries with it less and less of the initial restrictions I started with. By this point, I'm really not sure if I could be said to belong to ANY philosophy. Maybe logician. But probably not even that. | |
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| Philosophies of math...Includes essay...seriously
Posted: 11/8/2009 11:59:25 PM | | One and one is two. Right? After that it all meant for what??? Oh yes all of the rest that continues on and is only for the specialists who thrive on endless debate about NUMBERS. Keep going, I suppose, computers are also based on 1, 0, etc. And anyhoo...could you fix my car and measure my water pressure to account for bad pressure in my kitchen. Now that is useful math. Otherwise celebrate the multifold and exciting differences existing in all of the characters that are presently on earth. Open up. Constructionists, formalists, intuitionists, DANCE. Yep. | |
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| Philosophies of math...Includes essay...seriously
Posted: 11/9/2009 12:03:34 AM |
Sorry. I guess I'm just nowhere near as smart as you. Help a stupido out?
I can't use mathematical typeset on here, so it's easier to write it in english than to try using the shorthand. Most of it would be unreadable to most people unfamiliar with the symbology anyway, so what is the point?
It is not correct to say that english is too imprecise a means of expression, for what mathematical symbol has no english analogue? If they all do, and it is understood that an english expression is only "longhand" for the symbol, then what is the problem?
Maybe I should just accept that constructionivism is just something that I have to ignore?
A lot of people do.
I also don't know if the removal of trichotomy and irrational numbers is due to the removal of the law of the excluded middle, or down to something else.
If one denies the law of the excluded middle, the trichotomy law and irrational numbers fall like dominoes as a consequence. So why do the constructivists discard such a useful law?...Because, while they recognize its validity for finite sets, they feel it is an unjustified assumption when extended to the infinite.
I should say at this point that there are many schools of mathematical thought, but I have broken it into three primary divisions. My motive was to try to adhere to the KISS principle in acquainting people with the philosophy of mathematics, but in doing so, I have been "fibbing slightly. Strictly speaking, the constructivists and intuitionists are not exactly the same and in each case there are differing degrees of intuitionism and constructivism that we might consider them ranging from "soft core" to "hard core". There is also much overlap between them, so rather than confuse people with subtle distinctions, I thought it best to "lump them all together" as "constructivists." My justification for this imprecision was motivated by my desire to keep it simple and interesting for the non-mathematical readers of this thread, but it (sort of) has its roots in mathematics. As Babbage said to Tennyson: "In your otherwise beautiful poem, one verse reads:
Every moment dies a man,
Every moment one is born....
...I would suggest...Every moment 1 1/16 is born.
Strictly speaking, the actual figure is so long I cannot get it into a line, but I believe the figure 1 1/16 will be sufficiently accurate for poetry."
I don't know of ANY mathematicians who claimed that the set of integers includes "aleph null"
Frege & Russell's construction of the Naturals determined the cardinality of the set to be aleph null. So far as I know, that is the only part of the construction rejected by the constructivists. I imagine they would simply leave the cardinality as "indeterminate"
Same uni as Alan Turing's, FYI.
I guess his profs all retired by the time you started eh?
I haven't read much by Einstein that gives me the impression that he thought too deeply about either.
Quit reading about his science and start reading about the man. "Einstein, the Life and Times" is a good book to start with.
I haven't read that much that implies that he was all that compassionate either.
Here's a quote of his that I like:
"A human being is a part of the whole called by us universe, a part limited in time and space. He experiences himself, his thoughts and feeling as something separated from the rest, a kind of optical delusion of his consciousness. This delusion is a kind of prison for us, restricting us to our personal desires and to affection for a few persons nearest to us. Our task must be to free ourselves from this prison by widening our circle of compassion to embrace all living creatures and the whole of nature in its beauty." | |
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