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Emergent Properties

  1. Jun 4, 2005 #1
    Let us say that we have accepted the mathematical existence of lines. I then intersect them at a point and have a new thing I call an angle. In the example the lines are analogous to axioms and angle to an emergent property. The question is what is the existential status of the angle? Its existence depends upon the lines so it could not have existed before the axioms were set. If so how could something simply go from not existing to existing. Can we even talk of the angle not existing before the axioms were defined? Does the existence of the angle exist in the nature of a line and being able to arrange it? What this example also begs is what is meant by existent. Do we mean something that exists objectively or in some other way?

    I think the place to start is by asking where we got the idea for our lines [or whatever particular axioms we are considering]. To me it seems that the foundations of math must have their origin in experience. That is, the foundation of math is a posteriori, not a priori.
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  3. Jun 5, 2005 #2


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    Kant's idea that the basics of Euclidean geometry are a priori in the human mind is evidently wrong as stated, but neurologists and psychometricians have discovered that we do have some built-ins. New-born babies can respond based on simple arithmetic, although they surely have no conscious awareness of numbers, addition and subtraction.
  4. Jun 5, 2005 #3
    With all due respect you have not offered anything to the post. All you did was make two unsupported claims related to the last sentence and didn't even address the first paragraph. The idea of something's existence arising in dependence upon another is a idea out of Buddhist philosophy and not so much discussed in the West. At the same time, it has a relevance in math as my example has shown.
    Last edited: Jun 5, 2005
  5. Jun 8, 2005 #4


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    Why does it seem that way to you? As Adjoint points out, Kant was incorrect to say that the axioms of Euclidean geometry are innate to the human mind, but that alone does not mean that they derive from experience. Where in experience do you ever find a straight line, much less two straight lines that never intersect? The general concensus today is that mathematics is entirely analytic and a priori. Most math that I know of has empirical applications, but the logical foundations of the systems are not empirically derived.

    Let us consider Euclid's first postulate:

    For every point P and every point Q not equal to P there exists a unique line that passes through P and Q.

    This isn't true a posteriori. That is, the truth of this postulate is not contingent upon empirical investigation. It is true by virtue of the flat three-dimensional space in which Euclidean geometry resides. There is no conceivable flat three-dimensional world, without tears in the space, in which this postulate is not true. As such, the postulate is true because, in Euclid's system, it would be logically impossible for the postulate not to be true. That makes it an analytic, a priori truth.

    If you want to talk about the foundation of math, though, it would do you well to state what system of math you are talking about. Do you contend that all math is empirically derived? I would hope that I've just demonstrated that Euclidean geometry is not. My knowledge of mathematics is extremely limited, but I'm fairly certain that there are many other mathematical systems whose postulates, or axioms, are not empirically derived. Perhaps a more knowledgeable poster can come in and talk about a couple more.
  6. Jun 9, 2005 #5
    First, selfadjoint didn't prove anything. Second, who cares what the general consensus is. Third, you don't understand what I am talking about.

    First, what do you mean by the word "true"? In what sense is this true? This is not even clear. Are you using it in the sense that it is true that the earth rotates the sun? You are using these words w/o really understanding what you mean.

    Second, you don't understand what I am saying. I am not talking about the truth of the postulate being based on empirical evidence. The postulate and its truth is something that is the product of a conceptual mind. What I am saying is that where does the concept of straight come from? When we give direction we says "follow this road straight" and we know exactly what we mean. The mathematical abstraction of a straight line is no different than the abstraction from real people to the generality person.

    First, you haven't demonstrated anything. The point of my post is that the concepts that underly mathematics must come from experience. Math is a creation of the human mind. What is a point? It is nothing but an abstraction derived from experience. All geometry follows from there. You can not separate humans from their environment and expect to explain things. That is the downfall of older science. To understand humans you need to study them in their everyday life, not in an abstract scientific sense. Maybe you should read Heidegger's "Being and Time" since you seem to have so much advice for others.

    While both admire the beauty of their work, the difference between artists and mathematicians is that artists realize that they created their work while the mathematician doesn't.
  7. Jun 9, 2005 #6


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    That may or may not be true, but I certainly understand what is meant by the words "the foundation of math is a posteriori." That proposition has a long history that predates anything you've come up with yourself and is considered by almost everyone to be a false proposition. You'll need to come up with a better argument than you have thus far to swing concensus the other way.

    It's extremely clear and defined by truth tables. Given a propositional function, when you plug in the value "true," you get certain results that are different from when you plug in the value "false." That is all the word means here. Whether or not the proposition conforms to anything in empirical reality is a different matter and not important since math is analytic. Mathematical statements are true by definition alone, not because they conform to anything found in empirical reality.

    Perhaps that's what you meant to say, but that isn't what you said. You said that math was foundationally a posteriori. That statement entails that mathematical postulates are contingently true, true because they conform to what is found in empirical reality. That is not the case.

    Either way, I would contest that the concept of a perfectly straight line is derived from experience. As I asked before, where in experience do you ever find a perfectly straight line? When we tell a person to go straight down a given highway, that means to simply follow the road without turning. The road, however, is never a perfectly straight line. 'Straight' is here being used colloquially and doesn't mean the same thing that it means in Euclidean geometry.

    How so? You are making bald assertions that go against what is accepted by contemporary mathematicians and philosophers, but providing no argument to back up your assertions. Why is it that you think a point is an abstraction derived from experience?

    What does any of this have to do with mathematics?

    How so? If mathematical systems are entirely analytic, that means that they are entirely the creation of human minds. If math were empirical, that would not be the case. If math were empirical, then mathematical truths would be true by virtue of the universe we live in and would not be the creation of human minds. You don't even seem to be drawing the same conclusion that you initially drew now.
  8. Jun 9, 2005 #7


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    By the way, tensor, if you continue to contend that everyone responding to you does not understand what you say, you might want to consider the possibility that the failure lies in your explanation and not in our understanding. Perhaps you aren't actually saying what it is that you mean to say. You don't need to be so combative. This is math we're talking about, not politics. Don't even try to tell me you have an emotional investment in mathematics.
  9. Jun 9, 2005 #8
    Who cares what other people have come up with. My ideas come from my thinking. If all you can do to counter what I have said is refer to other thinkers then I nothing more to say.

    They are true in the sense that it is true that one should stop at a red light. They are nothing more than rules to follow. You present your axioms, define some rules and as a result derive further properties.

    Mathematical statements are contingently true. This is what I have been saying all along. There are only two ways mathematical statements can be true (1) They conform to empirical reality or (2) they conform to a set of rules. If you think there is another way to demonstrate the truth of mathematics then present one. If all you are going to say is that I am going against historical thought then I don't think you are on the same level of discussion as me. If you must use sources, Wittgenstein went against the traditional views of math and he is certainly well respected. But I really don't care what he thought.

    How do you conceive of the perfect woman? How do you conceive of the perfect job? You conceptually have an abstract image of what these things would be based on your interaction with a real woman or job. It is exactly the same with a line. Look how long mathematicians debated the reality of non-euclidean geometry. They could not conceive of it because it was so far removed from experience. If empirical experience didn't set the development of math then why was euclidean geometry done before non-euclidian? If these developments do not depend upon experience then you must come up with a reason why they were discovered in this order.

    There you go again telling me what other people think. I don't care. Defeat my arguments with your own logic and reason. Because a point is derived from the idea of location. Jim is over there. The car is over there. We abstract "over there" from the empirical world to the mathematical world of a point.

    Think about it.

    The truths are a creation of the mind and whatever the mind can conceive is based upon experience. There is no a priori understanding of mathematics. Even the idea of a flying pink elephant although the object of the conception does not exist the mind does exist. What is the mind conceiving? Even though there is no elephant with these qualities (1) pink is a color (2) other animals can fly and (3) elephants exist. There is nothing we can conceive that is not based upon experience. If you don't agree then give an example.
  10. Jun 9, 2005 #9
    I only said that you don't understand.

    Are you now going to tell me that philosophers and mathematicians in the past didn't have an emotional investment in math?
  11. Jun 10, 2005 #10


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    Not all mathematical statements are contingent; Some are always true, some always false. Of course, it depends on which system you're talking about, but this is the case in the most common systems. In math, truth is not determined by conformity to reality; Truth is determined by the rules of the system. Mathematicians get to create the rules, and most, if not all, are well aware of this.
    Well, where are you cutting off the relationship? You could say that the concept of whole numbers were abstracted from experience by counting finite numbers of real objects. But would you still say the concept of the infinite set of whole numbers is also abstracted from experience?
  12. Jun 10, 2005 #11
    You have basically restarted what I have stated. Even those established as true are contingent though. they are contingent on conforming to the rules upon which they were established. The truths of mathematics are no different than the truths of Shoots and Ladders. As for any evidence to back up your claim that mathematical truths are not dependent on empirical truths you provided no evidence.

    Sure. If you can conceive of doing something once then twice and you have an understanding of this activity comming to an end you also have the concept of it occuring indefinitely.
  13. Jun 10, 2005 #12


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    No, I plainly contradicted what you said.
    The "establishing as true" is either defining or proving. That's all you do in math: define and prove. And you define what is true. "Truth" is just a function, a set. That's it.
    Again, not all statements are contingent. Tautologies and contradictions are non-contingent statements. You can look this up elsewhere if you don't believe me.
    Are you just making a distinction between axioms and theorems? Or something else?
    What are the truths of Shoots and Ladders?
    I said, "In math, truth is not determined by conformity to reality; Truth is determined by the rules of the system." That is just how math works. I can explain it to you if you want. Or if you don't trust me, you can learn about math from somewhere else.
    But I have never experienced anything infinite. What you just described is an abstraction from an abstraction. There is another level there. The infinite set is an abstract object. The numbers are also abstract objects. Rules and definitions are abstract objects. Math is based on and deals with abstract objects.

    Also, are you suggesting you know of an a posteriori proof that two parallel straight lines -lines are infinitely long- never intersect? Maybe I've misunderstood- a posteriori proof is just experience, right? How do you observe a one-dimensional object? And an infinite length of space? And make sure it is actually percfectly straight?

    The only thing I can see going for you is that most humans happen to be born with working senses but not a working knowledge of math. Can you argue that it's impossible for a blind human to understand mathematics? Or a deaf human? ...
    Last edited: Jun 11, 2005
  14. Jun 11, 2005 #13
    You words may have contradicted what I said but if you look deeper you will see that you agreed with my main point. I said they are contingently true (which you disagree with) and can be so in only two ways (1) conformity to reality (2) conformity to a set of rules. In either case the truths are contingently true. Since you said that "truth is determined by the rules of the system" you implicitly agree with my categorization and therefore the contingent nature of mathematical truths.

    If the best you can do is refer me to authority then we have no room left to talk. Why should I belive you. The only correct thing you have said so far was a repetition of what I already wrote.

    The rules of the game. As you put it "truth is determined by the rules of the system". If I land on a space that have a shoot I must go down. This is a truth of the game. If I land on that space and move up I have contradicted the rules and it is false.

    Where I come from we provide reasons for our statements. If the best you can do is say "because that is how it is" you need to go to the kindergarden message board. In addition, I have two university degrees in physics and math so I don't think you need to teach me about math but maybe I could teach you shoots and ladders since its a little kid's game.

    You haven't made a contestable statement here. Think about walking. We can all conceive of walking and as long as we don't bump into something we keep going. This is an understanding of infinity. What I was saying re: the flying pink elephant is that all our concepts come from experience with the empirical world. There is no human nature or innate human abilities that are simply there as a result of being born. There is only experience and habit.

    I don't really care about your opinions. Make contestable statements. I have made several assertions that you have not refuted. That is I make a claim and you need to find an example that contradics my position.
  15. Jun 11, 2005 #14
    Topology and geometry are abstract as are space, distance and position. Structures are easy to visualize. However since they are abstract they cannot be used to explain physical processes.
    Last edited: Jun 11, 2005
  16. Jun 11, 2005 #15
    Just say no to drugs.
  17. Jun 11, 2005 #16


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    Okay, I can better explain what I meant. (Classical) Propositional logic is simple and the system I know the most about, so I'll stick with it. A valuation on formal language L is a function from the set of formulas of L to a set of two distinct objects. A formula f is contingent iff there exist valuations V and W such that the value that V assigns to f does not equal the value that W assigns to f. That's what I meant by 'contingent'. How else do you evaluate a formula (mathematical statement)? Is there even any reason to interpret x as true and y as false (or x as false and y as true)?
    Since most people usually use true and false instead of x and y, by truth being determined by the rules, I just meant part of the system is a valuation. The valuation is not reality; It's an abstract object. I guess you could try to define the valuations to conform to reality, but I don't know how you would get around the subjective nature of experience.
    You're talking about things that have already been precisely defined. If you're confused about the definitions, what else can you do but look them up? Are you talking about math or about your own original ideas?
    "Tautologies and contradictions are non-contingent statements," is true by definition.
    Why do the inference rules have truth-values? Wouldn't the formulas of the game be things like "Player p is on space x at turn t" or "Player p lands on space x implies player p moves to space y"? Also, how do you work out the valuation?
    V(f) = {x iff everyone playing the game agrees that f is true, y otherwise}
    V(f) = {x iff every player observes f, y otherwise}
    ? The truth of "Player p lands on space x implies player p moves to space y" would then depend on whether the players actually followed the rules of the game, if and what they agree they observed, or, worse, people having access to each other's conscious experience. That's probably why mathematicians define valuations instead of letting them depend on reality.
    I have some advice too: Read PF's rules. Insults aren't allowed. Luckily for me, the number of degrees you've earned doesn't determine the truth of your arguments. You didn't learn about logical fallacies in your math classes?
    How do babies learn how to breath? They inhale fluid in the womb. They do many other things in the womb before they even begin to store memories. How do they learn to do those things? What have you read about child development that makes you think humans have no innate abilities?
    Besides, I never said people were born with a knowledge of math. In fact, I said, "most humans happen to be born with working senses but not a working knowledge of math."
    If you don't care about my opinions, then you don't need to read them. I don't have to refute all of your assertions, and I wasn't trying to. If you explained your position as much as you changed the subject and insulted people, it would be easier to understand what exactly your position is. You're the one making claims that you haven't supported.
    Opinions are contestable statements, so what are you talking about?

    You may not have seen something I added:
    Also, are you suggesting you know of an a posteriori proof that two parallel straight lines -lines are infinitely long- never intersect? Maybe I've misunderstood- a posteriori proof is just experience, right? How do you observe a one-dimensional object? And an infinite length of space? And make sure it is actually percfectly straight?
  18. Jun 11, 2005 #17
    The original post was looking at where the emergent properties in math come from. We took as an example of axioms two lines and we intersected them. They form an angle which is an emergent property in that it depends upon the lines. The question was when does the angle come into existence. If it depends upon the lines then it could not exist before the lines existed. How could this thing called an angle suddenly pop into existence? The discovery of the angle did not uncover something that existed. It was merely a by-product of our axioms and certain operations defined.
    Last edited: Jun 11, 2005
  19. Jun 12, 2005 #18


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    (Just for the record) A mathematical structure S consists of:
    1) A non-empty set U, the universe or domain of S; The members of U are called the individuals of S;
    2) A set of basic operations on U;
    3) A non-empty set of basic relations on U.
    (ex. for set theory, individuals are sets, no basic operations, basic relations are identity and membership.)

    I'm just learning this, but I think you just create a formal language L, an interpretation (a mathematical structure tailored to L), and a valuation. Of course, you also have a metatheory where you prove everything about the object theory. Anyway, you can then combine the basic things of your structure to produce other "emergent" things. Call the basic things of your structure "ingredients", the emergent things "products", all the rest "rules", and the process of producing products "production". :rolleyes:
    So the question is what is necessary and/or sufficient for what?

    The rules and set of ingredients are all that is used to produce the products, so they must be sufficient for the products. I think the rules and set of ingredients are necessary for the products. If so, the rules and set of ingredients, together, would be equal (or equivalent) to the products. You can also make it so each individual ingredient and rule is necessary for the products (just don't use unecessary ones). But even when combined with the rules, I don't think the individual ingredients are sufficient for the products; The individual ingredients must be combined with each other in order to produce the products. For example, you can combine sets, membership, and rules to produce subsets. I think your example would be the line, intersection, and rules combining to produce the angle. Does that make sense, sound right?

    I mean for "x is necessary for y" = "y implies x" and "x is sufficient for y" = "x implies y". But I'm not sure if that extends to "x depends on y", "x exists before y", etc.

    I need to look more at the whole setup and argument before I can clear some of that up, decide if it should be equal or equivalent, etc. But I don't have time just now. I'll just add a couple questions I'm thinking about: Are there different ways of producing the same products? That is, in the same system, can you use different rules and ingredients to produce the same products? Given that there are often several different proofs of the same theorem, it seems possible, though proofs are a bit different. Can ingredients and products be switched? That is, can you use the products of one system as the ingredients of another system and produce the same products (the ingredients of the other system)?
  20. Jun 12, 2005 #19


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    People use abstract objects to explain- or try to explain- physical processes; Think of scientific theories. Do you mean they cannot model or explain reality perfectly or with absolute certainty?
  21. Jun 14, 2005 #20
    They can and they have done it well until now. From now on (within the next 100 years or so) i believe something revolutionary will happen.

    Thomas Kuhn writes how new ideas and new ways of thinking displace the old. He invented the term 'paradigm shift' to describe what happens when 'normal science' runs into 'anomalies' and enters a 'crisis', which in turn leads to a 'scientific revolution'.
    Last edited: Jun 14, 2005
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