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The misuse of science by William Lane Craig and othe Christian apologists.

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  • Originally posted by shunyadragon View Post
    The line or distance between two points A and B can be divided an infinite number of times, as an actual infinite set, and possibly an infinite number of points. The points have no dimensions.

    You are conflating an actual infinity with a potential infinity. The actual infinity of dividing the line A and B is not a potential infinity.
    Actually, I think you're the one that is confused, here.

    The process of continuously dividing a line into smaller and smaller sections is "potentially infinite." It is an iterative process which can be iterated indefinitely.

    The number of points which compose the line, on modern geometry, is "actually infinite." That is, there are an infinite number of points which coexist together in the object which we call a line.

    Hansgeorg seems to reject all number theory which was developed after about 100 CE, so it's not really surprising that he would reject the idea of actual infinites.
    "[Mathematics] is the revealer of every genuine truth, for it knows every hidden secret, and bears the key to every subtlety of letters; whoever, then, has the effrontery to pursue physics while neglecting mathematics should know from the start he will never make his entry through the portals of wisdom."
    --Thomas Bradwardine, De Continuo (c. 1325)

    Comment


    • Originally posted by Boxing Pythagoras View Post
      Yes, this is the manner of counting learned by kindergartners.
      As you are a teacher, you risk to overvalue the education system.

      What you call the manner of counting learned by kindergartners is one major study of Boethius and Pythagoras, it is known as Arithmetic.

      By the way, parts of it are even advanced mathematics today, like prime number theory.

      Originally posted by Boxing Pythagoras View Post
      By the time you get to fourth grade, you'll start learning about Rational numbers.
      No longer numbers, but proportions. We are dealing with what Boethius and Euclid called Geometry - or at least with what Pythagoras called Music.

      Originally posted by Boxing Pythagoras View Post
      By seventh grade, you'll have some experience with the Irrationals.
      That sets you diagonally and squarely into the realm of geometry.

      Originally posted by Boxing Pythagoras View Post
      In high school, you'll start to learn about the Complex numbers. And, should you pursue a degree in mathematics, you might have the opportunity to study set theory and non-standard analysis, which will introduce you to infinite numbers.
      Here we are no longer dealing with number or proportion or anything else as abstractable aspects of reality, but with pure fictions.

      Originally posted by Boxing Pythagoras View Post
      I don't. Which is why I have been consistently stating that infinity is not a number.
      Well, that means you are not calling your Hyperreal numbers numbers.

      One good thing.

      Originally posted by Boxing Pythagoras View Post
      It's not ad hoc, in the slightest. If element771 had said, "Red divided by 2 is still red. Half of red is still red," then it would not be ad hoc for me to point out that "red" is not a number, right? Neither is it ad hoc to point out that element771 was attempting to perform numerical operations on things which are not numbers, in his post.
      I agree that "infinity" is not mathematical any more than any number or size is infinite.

      Originally posted by Boxing Pythagoras View Post
      I've been utilizing the Hyperreals in my example, so I'll give the definition utilized in that number system. A number, X, is infinite when |X|>|r| for any Real number, r.
      That definition cannot have a referent.

      Originally posted by Boxing Pythagoras View Post
      You do realize that books intended for schoolchildren do not encompass the sum totality of the philosophy of mathematics, right? Such textbooks almost never even provide a rigorous treatment of the subject matter which they do teach, let alone of mathematics in general. The fact that your schoolbook utilizes a particular pedagogical method tells us nothing at all about how numbers are defined.
      The French schoolbooks have taken up the position from a mathematician who did some rigorous study of the concept.

      Originally posted by Boxing Pythagoras View Post
      Is it possible to define 4 without reference to any other numbers? If not, then why are you making an arbitrary distinction between the Natural numbers and other numbers which are defined by reference to other numbers?
      I did not say "relative numbers are defined BY reference to other numbers".

      I did say words to the effect as relative numbers are defined as the reference of one natural number to another natural number. That would include the "one more than" which is a standard ingredient in the definition of any number.

      Originally posted by Boxing Pythagoras View Post
      You said that 4 is defined as "1 more than 3." So, 4 by definition is a comparison between two numbers. Since it is necessary to define a number before that number can have any meaning, it seems that even upon your definition, 4 is a relative number.
      No, in this definition, "4 is 1 more than 3" there is exactly one thing which IS a comparison, namely "1 more than". As to 4, it IS not a comparison, it is KNOWN BY a comparison, namely to a known number, 3.

      Not all numbers that are COMPARED are relative numbers, only those that EXPRESS THE FACT OF COMPARING are so.

      > and < express comparison.

      =1+, =2+, =3+ (or =n+1, =n+2, =n+3) and so on explicitate > and are positive relative numbers.

      =n-1, =n-2, =n-3 and so on explicitate < and are negative relative numbers.

      = also expresses comparison and is synonymous to ±0 which is a relative number.

      Originally posted by Boxing Pythagoras View Post
      That'd be all well and good if we were discussing an obvious fact. We are not.
      You seem to have lost grasp of obvious facts somewhere in high school or university.

      Originally posted by Boxing Pythagoras View Post
      Ahh, so then you do agree that there are places in which Aristotle, and Aquinas after him, were incorrect in their reasoning?
      Aquinas was incorrect in what he presumed to be good astronomy, but very correct in reasoning that the visible phenomena reducible to Ptolemaic orbits could be reducible to some other kind of orbits - as they are to Tychonians ones.

      Originally posted by Boxing Pythagoras View Post
      Infinity is not a number. Mathematical operations are only defined for numbers. "Infinity minus [any number]" does not equal infinity, because the statement is not even cogent. It is just as nonsensical as the statement "blue minus any number" or "deliciousness minus any number" or "hansgeorg minus any number."
      Thank you for taking quality and substance as parallels to "infinity".

      The qualities of "being", "good", "wise", "eternal" all express a non-mathematical infinity attributable to God.

      And "infinite substance" is a definition of God.

      Originally posted by Boxing Pythagoras View Post
      However, there are numbers which have the property of being infinite. If you subtract any finite number from an infinite number, you do find that the difference is also an infinite number. However, it is not necessarily the same infinite number. The fact that two numbers are infinite does not imply that they are equal any more than the fact that two numbers are even implies that they are equal.
      Subtracting is an operation and therefore is not defined for infinity.

      You have not given any indication that a number can be infinite. You have only asserted it.

      Originally posted by Boxing Pythagoras View Post
      And I am happy to disagree with Cantor on this point. It is incredibly easy to define a Hyperreal number which is equal to the quantity of Natural numbers, and another Hyperreal which is equal to the quantity of only the even Natural numbers, and to compare these two numbers.
      If you disagree with Cantor, you disagree with the man who came up with tha crackpot idea of making natural numbers a collected set.

      Originally posted by Boxing Pythagoras View Post
      No, the Real numbers actually do deal with numbers. One of the most famous of all the Real numbers, Euler's number, is not defined in terms of geometry, at all. In fact, the only way to rigorously define the very concept of the Reals is with sets of numbers.
      The last first : what you call a rigorous definition is probably a wrong one. Not wrong as in one which would give false results, but as in one which is philosophically off.

      Now, as sqrt of 2 has a geometric definition involving a square and its diagonal and sides, as pi has a definition of a circumference compared to a diagonal (which remains the same across all pairs of positions that are defined by its crossing the centre, or it wouldn't be that of a circle), so also Euler's quantity is a comparison of magnitudes rather than of numbers.

      You have seen graphs with exponential growth? You have seen slides?

      When points "0" and "1" and "2" are united by 301 (0,1), 477 (0,2) and 176 (1,2) units, there is certainly not just some 10-log arithmetically defined behind this (even while 0.301 is a fair approximation of 10-log of 2 etc.), but there is also some Euler's quantity involved. And Euler's quantity is certainly rather a comparison of magnitudes than a natural number.

      Originally posted by Boxing Pythagoras View Post
      Naming conventions are fairly irrelevant to the applicability of mathematics to the real world. We could just as easily have named these "Totally-Applicable-to-the-Real-World numbers" instead of "Surreal numbers." Or "banana dog numbers." Or "letters of the alphanumbers." Or "mxyzptlk." Whether or not these numbers are applicable to the real world is not dependent upon their name.
      However, I very much suspect that the one who named them was not counting on a very straightforward applicability.

      Originally posted by Boxing Pythagoras View Post
      Now you are attempting to draw a distinction between quantity and number? So, you would say that zero represents an actual quantity, but not an actual number?
      Euler had "quantity" as most common concept, subdivided into number and magnitude.

      In arithmetic, 0 is not a number, but it exists as a relative number. In geometry, 0 is not a magnitude, but a position surrounding certain magnitudes, and often one arbitrarily chosen.

      Originally posted by Boxing Pythagoras View Post
      Thus far in the discussion we have maintained that we are not taking a Platonist view of the ontology of numbers. As such, all numbers-- including the Natural numbers-- only have "imagined and therefore conventional existence." Are you trying to claim that the Natural numbers exist in the Platonist sense, but all other numbers exist only in the Nominalist sense?
      I am unlike some others - and I think I already stated this - a Platonist about 1, 2, 3 or sqrt of 2, pi, e, phi.

      The other position outlined is not nominalist but fictionalist.

      I am definitely a fictionalist about i, but also about "-3" taken as if not a relative number.
      http://notontimsblogroundhere.blogspot.fr/p/apologetics-section.html

      Thanks, Sparko, for telling how I add the link here!

      Comment


      • Originally posted by Boxing Pythagoras View Post
        The number of points which compose the line, on modern geometry, is "actually infinite." That is, there are an infinite number of points which coexist together in the object which we call a line.

        Hansgeorg seems to reject all number theory which was developed after about 100 CE, so it's not really surprising that he would reject the idea of actual infinites.
        I am rejecting the idea of a line being composed of a number of points.

        A line can potentially be divided into any number of points.

        I am NOT rejecting all number theory developed after 100 AD, I am rejecting the changes of definitions to already defined concepts.

        For instance, I do not deny logarithms, I just say they do not really, only fictionally, involve fractional or real number exponents. They must really be about sth else, like geometry.

        Or I do not deny prime number theorems developed in 20th C, nor topology as a very abstract and quantitatively less defined branch of geometry.
        http://notontimsblogroundhere.blogspot.fr/p/apologetics-section.html

        Thanks, Sparko, for telling how I add the link here!

        Comment


        • Originally posted by shunyadragon View Post
          The points have no dimensions.
          That is why they only exist as limits between lines.

          Precisely as a line not having two dimensions only exists as a limit between surfaces.

          Precisely as a surface, not having three dimensions of concrete physical existence only exist as limits between bodies.
          http://notontimsblogroundhere.blogspot.fr/p/apologetics-section.html

          Thanks, Sparko, for telling how I add the link here!

          Comment


          • Originally posted by hansgeorg View Post
            By the way, parts of it are even advanced mathematics today, like prime number theory.
            Do you realize that the Prime Number Theorem is built upon mathematics using the Real numbers and infinite summation? Aren't these things you reject? It's not like the Prime Number Theorem is based solely upon one's ability to list the Natural numbers.

            Here we are no longer dealing with number or proportion or anything else as abstractable aspects of reality, but with pure fictions.
            Considering the fact that you reject all modern physics and mathematics for at least 700 years, I doubt you'll find this very convincing, but the Complex numbers are utilized in physics every day in order to represent aspects of reality.

            Well, that means you are not calling your Hyperreal numbers numbers.
            No, it means I'm not calling "infinity" a number. The Hyperreal numbers are, as their name implies, numbers. Those numbers can be infinite despite the fact that infinity is not a number in exactly the same way that a number can be prime despite the fact that primeness is not a number.

            I agree that "infinity" is not mathematical any more than any number or size is infinite.
            I never said infinity is "not mathematical." I said it is not a number. A number can be infinite, but infinity is not a number.

            That definition cannot have a referent.
            Sure it can. Mathematics on infinite Hyperreal numbers is perfectly consistent.

            The French schoolbooks have taken up the position from a mathematician who did some rigorous study of the concept.
            If you can present me some work by this mathematician in which he states that only the Natural numbers are numbers and that no other mathematical objects bearing the name "number" are actually numbers, I'd be happy to respond to his arguments. Otherwise, as I said, the pedagogical methods employed by schoolbooks intended for children hardly form the totality of mathematics.

            Aquinas was incorrect in what he presumed to be good astronomy
            Great! So we are in agreement, then, that Aquinas can be incorrect. As such, appealing to his authority is a fairly useless venture. If you are convinced by some argument he has made, then absolutely present it and we will discuss that argument. Don't simply cite Aquinas as if his word constitutes reality.

            Subtracting is an operation and therefore is not defined for infinity.
            Well, then, it's a good thing I wasn't performing that operation on infinity, but rather on numbers which are infinite; just as I cannot perform subtraction on primeness, but rather on numbers which are prime.

            You have not given any indication that a number can be infinite. You have only asserted it.
            ...and I've discussed a field of mathematics in which infinite numbers are utilized perfectly consistently. As such, if you want to claim that numbers cannot be infinite, either you need to show some sort of inconsistency which results from the use of Hyperreal numbers or else you need to demonstrate that Hyperreal numbers cannot be applicable to the real world. You have done neither of these.

            If you disagree with Cantor, you disagree with the man who came up with tha crackpot idea of making natural numbers a collected set.
            Firstly, Cantor was not "the man who came up with tha [sic] crackpot idea of making natural numbers a collected set." Philosophers of mathematics had been discussing the question of whether or not the Natural numbers could be considered a set for some time before Cantor. His contribution was to provide a set of rules by which infinite sets could be discussed and explored in a meaningful way.

            Secondly, who cares that I may disagree with him on some point? In point of fact, I disagree with him in many places-- as did the mathematicians who continued his work. Modern Set Theory is very different than the original idea which Cantor proposed precisely because of those disagreements. That doesn't mean that actually infinite sets cannot exist in the real world.

            The last first : what you call a rigorous definition is probably a wrong one. Not wrong as in one which would give false results, but as in one which is philosophically off.
            What is "philosophically off" about the mathematical definition of the Real numbers?

            When points "0" and "1" and "2" are united by 301 (0,1), 477 (0,2) and 176 (1,2) units, there is certainly not just some 10-log arithmetically defined behind this (even while 0.301 is a fair approximation of 10-log of 2 etc.), but there is also some Euler's quantity involved. And Euler's quantity is certainly rather a comparison of magnitudes than a natural number.
            I never claimed that Euler's constant is a Natural number. To do so would be fairly obviously silly. Rather, I stated that the definition of e is not derived from geometry.

            However, I very much suspect that the one who named them was not counting on a very straightforward applicability.
            Actually, in both cases, they were. Abraham Robinson developed the Hyperreal number system for use in non-standard analysis. John Conway developed the Surreals while working on applications of game theory to the board game, Go.

            Euler had "quantity" as most common concept, subdivided into number and magnitude.

            In arithmetic, 0 is not a number, but it exists as a relative number. In geometry, 0 is not a magnitude, but a position surrounding certain magnitudes, and often one arbitrarily chosen.
            So, now you're saying that zero is not a quantity. Some consistency would be nice.
            "[Mathematics] is the revealer of every genuine truth, for it knows every hidden secret, and bears the key to every subtlety of letters; whoever, then, has the effrontery to pursue physics while neglecting mathematics should know from the start he will never make his entry through the portals of wisdom."
            --Thomas Bradwardine, De Continuo (c. 1325)

            Comment


            • Originally posted by Boxing Pythagoras View Post
              Considering the fact that you reject all modern physics and mathematics for at least 700 years, I doubt you'll find this very convincing, but the Complex numbers are utilized in physics every day in order to represent aspects of reality.
              FWIW, in a course on electronics I had, we used them to model impedances, voltages, electrical currents, etc. in AC circuits.
              We are therefore Christ's ambassadors, as though God were making his appeal through us. We implore on Christ's behalf: 'Be reconciled to God!!'
              - 2 Corinthians 5:20.
              In deviantArt: ll-bisto-ll.deviantart.com
              Christian art and more: Christians.deviantart.com

              Comment


              • Originally posted by Boxing Pythagoras View Post
                Do you realize that the Prime Number Theorem is built upon mathematics using the Real numbers and infinite summation? Aren't these things you reject? It's not like the Prime Number Theorem is based solely upon one's ability to list the Natural numbers.
                I would be surprised if you and hansgeorg mean the same thing by "prime number theory".
                Jorge: Functional Complex Information is INFORMATION that is complex and functional.

                MM: First of all, the Bible is a fixed document.
                MM on covid-19: We're talking about an illness with a better than 99.9% rate of survival.

                seer: I believe that so called 'compassion' [for starving Palestinian kids] maybe a cover for anti Semitism, ...

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                • Originally posted by hansgeorg View Post
                  He very clearly does claim physical existence if most likely finite. He concludes existence of God from that.

                  THEN he blunders about attributes of God and makes the universe share His eternity, because he didn't realise how God could ever be motivated to create. Hence God moving it around Earth each day (yes, the classic Aristotelian and Thomistic prima via proof of God is a geocentric one) cannot be attributed to God's activity, but in his final analysis only to the Universe's activity from love of God, and thus to God as a final rather than strictly efficient cause. Hence the coeternity of Universe with God in late Aristotle and markedly upheld by St Thomas Aquinas' Averroistic opponents as not just a philosophic possibility but actual fact.
                  A few days' worth of posts were lost in the recent crash of the TWeb site. Suffice it to say, hansgeorg was not able to provide any texts of Aristotle proving hansgeorg's contention that Aristotle clearly claims physical existence is most likely finite. He admitted that he was only making assumptions based on diverse views of a couple of medieval Christian theologians and not on any texts of Aristotle.
                  βλέπομεν γὰρ ἄρτι δι᾿ ἐσόπτρου ἐν αἰνίγματι, τότε δὲ πρόσωπον πρὸς πρόσωπον·
                  ἄρτι γινώσκω ἐκ μέρους, τότε δὲ ἐπιγνώσομαι καθὼς καὶ ἐπεγνώσθην.

                  אָכֵ֕ן אַתָּ֖ה אֵ֣ל מִסְתַּתֵּ֑ר אֱלֹהֵ֥י יִשְׂרָאֵ֖ל מוֹשִֽׁיעַ׃

                  Comment


                  • I think we lost several posts during the recent site downtime, so rather than rehash the discussion, I'll try to pose the questions which I think remained open (or, at least, the questions whose answers I did not see) when TWeb went down.

                    So, hansgeorg:

                    Where is it that you think Aristotle has discussed the things which I have claimed are actual infinites, and where does he show that these things are instead potential infinites? It does not seem to appear in Physics, which is where Aristotle discusses whether a continuum can be composed of indivisibles and where he addresses Zeno's paradoxes of motion.

                    Where is it that you think Aristotle claims a point only exists in relation to a line, a line only exists in relation to a plane, and a plane only exists in relation to a three-dimensional space? Again, this is nowhere in Physics.

                    Where is it that you think Aristotle claims that the cosmos is either spatially or temporally finite? This seems entirely contrary to what Aristotle explicitly claims in Physics.
                    "[Mathematics] is the revealer of every genuine truth, for it knows every hidden secret, and bears the key to every subtlety of letters; whoever, then, has the effrontery to pursue physics while neglecting mathematics should know from the start he will never make his entry through the portals of wisdom."
                    --Thomas Bradwardine, De Continuo (c. 1325)

                    Comment


                    • Originally posted by Roy View Post
                      I would be surprised if you and hansgeorg mean the same thing by "prime number theory".
                      I would be surprised if hansgeorg means the same by the word "is" as everyone else does by the word "is".

                      Comment


                      • Originally posted by Chrawnus View Post
                        I would be surprised if hansgeorg means the same by the word "is" as everyone else does by the word "is".
                        Agree!
                        Glendower: I can call spirits from the vasty deep.
                        Hotspur: Why, so can I, or so can any man;
                        But will they come when you do call for them? Shakespeare’s Henry IV, Part 1, Act III:

                        go with the flow the river knows . . .

                        Frank

                        I do not know, therefore everything is in pencil.

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