No rigorous proofs from 200 B.C to 1870?











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15
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I'm reading: Mathematical thought from ancient to modern times by Kline. My question is about this pasasge:




Beyond its achievements in subject matter, the nineteenth century
reintroduced rigorous proof. No matter what individual mathematicians may
have thought about the soundness of their results, the fact is that from about
200 B.C. to about 1870 almost all of mathematics rested on an empirical and
pragmatic basis. The concept of deductive proof from explicit axioms had
been lost sight of. It is one of the astonishing revelations of the history of
mathematics that this ideal of the subject was, in effect, ignored during the
two thousand years in which its content expanded so extensively.




I wasn't aware of this at all. To me mathematics is about rigour, so to hear that this rigour is relatively new to mathematics suprised me. Mathematics that is based on emperical and pragmatic basis seems more like applied mathematics to me, not as pure mathematics.



So in that sense, can I see the mathematics from 200 B.C. to 1870 as mainly applied mathematics ? And the pure mathematics as I know it, is this mainly created in the last $pm$150 years ?










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  • 7




    @abel: "Elements" is dated c. 300BC. Note that Kline says "reintroduced", not "introduced".
    – user64687
    Jan 24 '15 at 23:28






  • 6




    i think the word "rigour" as it relates proof is subjective. today some may say that some of of euler's proof are not rigorous.
    – abel
    Jan 24 '15 at 23:31








  • 6




    I guess he wouldn't consider Cauchy (d. 1857) or Boole (d. 1864) rigorous? That's a pretty bold claim.
    – charlotte
    Jan 24 '15 at 23:32






  • 2




    @charlotte: Cauchy is an interesting example. He tried to be rigorous, but he didn't quite get there! For instance, he "proved" that a pointwise limit of a sequence of continuous functions is continuous. On the other hand, his approach certainly wasn't "empirical" or "pragmatic" as Kline suggests.
    – user64687
    Jan 24 '15 at 23:36








  • 2




    I think "empirical" is a bit of sloppy use of language. Euler's product formula for the zeta function is an example. This was shown by Euler, but using techniques that did not have a rigorous foundation. But it wasn't merely observation - Euler made an argument, it just wasn't a complete argument.
    – Thomas Andrews
    Jan 24 '15 at 23:36















up vote
15
down vote

favorite
3












I'm reading: Mathematical thought from ancient to modern times by Kline. My question is about this pasasge:




Beyond its achievements in subject matter, the nineteenth century
reintroduced rigorous proof. No matter what individual mathematicians may
have thought about the soundness of their results, the fact is that from about
200 B.C. to about 1870 almost all of mathematics rested on an empirical and
pragmatic basis. The concept of deductive proof from explicit axioms had
been lost sight of. It is one of the astonishing revelations of the history of
mathematics that this ideal of the subject was, in effect, ignored during the
two thousand years in which its content expanded so extensively.




I wasn't aware of this at all. To me mathematics is about rigour, so to hear that this rigour is relatively new to mathematics suprised me. Mathematics that is based on emperical and pragmatic basis seems more like applied mathematics to me, not as pure mathematics.



So in that sense, can I see the mathematics from 200 B.C. to 1870 as mainly applied mathematics ? And the pure mathematics as I know it, is this mainly created in the last $pm$150 years ?










share|cite|improve this question


















  • 7




    @abel: "Elements" is dated c. 300BC. Note that Kline says "reintroduced", not "introduced".
    – user64687
    Jan 24 '15 at 23:28






  • 6




    i think the word "rigour" as it relates proof is subjective. today some may say that some of of euler's proof are not rigorous.
    – abel
    Jan 24 '15 at 23:31








  • 6




    I guess he wouldn't consider Cauchy (d. 1857) or Boole (d. 1864) rigorous? That's a pretty bold claim.
    – charlotte
    Jan 24 '15 at 23:32






  • 2




    @charlotte: Cauchy is an interesting example. He tried to be rigorous, but he didn't quite get there! For instance, he "proved" that a pointwise limit of a sequence of continuous functions is continuous. On the other hand, his approach certainly wasn't "empirical" or "pragmatic" as Kline suggests.
    – user64687
    Jan 24 '15 at 23:36








  • 2




    I think "empirical" is a bit of sloppy use of language. Euler's product formula for the zeta function is an example. This was shown by Euler, but using techniques that did not have a rigorous foundation. But it wasn't merely observation - Euler made an argument, it just wasn't a complete argument.
    – Thomas Andrews
    Jan 24 '15 at 23:36













up vote
15
down vote

favorite
3









up vote
15
down vote

favorite
3






3





I'm reading: Mathematical thought from ancient to modern times by Kline. My question is about this pasasge:




Beyond its achievements in subject matter, the nineteenth century
reintroduced rigorous proof. No matter what individual mathematicians may
have thought about the soundness of their results, the fact is that from about
200 B.C. to about 1870 almost all of mathematics rested on an empirical and
pragmatic basis. The concept of deductive proof from explicit axioms had
been lost sight of. It is one of the astonishing revelations of the history of
mathematics that this ideal of the subject was, in effect, ignored during the
two thousand years in which its content expanded so extensively.




I wasn't aware of this at all. To me mathematics is about rigour, so to hear that this rigour is relatively new to mathematics suprised me. Mathematics that is based on emperical and pragmatic basis seems more like applied mathematics to me, not as pure mathematics.



So in that sense, can I see the mathematics from 200 B.C. to 1870 as mainly applied mathematics ? And the pure mathematics as I know it, is this mainly created in the last $pm$150 years ?










share|cite|improve this question













I'm reading: Mathematical thought from ancient to modern times by Kline. My question is about this pasasge:




Beyond its achievements in subject matter, the nineteenth century
reintroduced rigorous proof. No matter what individual mathematicians may
have thought about the soundness of their results, the fact is that from about
200 B.C. to about 1870 almost all of mathematics rested on an empirical and
pragmatic basis. The concept of deductive proof from explicit axioms had
been lost sight of. It is one of the astonishing revelations of the history of
mathematics that this ideal of the subject was, in effect, ignored during the
two thousand years in which its content expanded so extensively.




I wasn't aware of this at all. To me mathematics is about rigour, so to hear that this rigour is relatively new to mathematics suprised me. Mathematics that is based on emperical and pragmatic basis seems more like applied mathematics to me, not as pure mathematics.



So in that sense, can I see the mathematics from 200 B.C. to 1870 as mainly applied mathematics ? And the pure mathematics as I know it, is this mainly created in the last $pm$150 years ?







math-history






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asked Jan 24 '15 at 23:22









Kasper

5,72884887




5,72884887








  • 7




    @abel: "Elements" is dated c. 300BC. Note that Kline says "reintroduced", not "introduced".
    – user64687
    Jan 24 '15 at 23:28






  • 6




    i think the word "rigour" as it relates proof is subjective. today some may say that some of of euler's proof are not rigorous.
    – abel
    Jan 24 '15 at 23:31








  • 6




    I guess he wouldn't consider Cauchy (d. 1857) or Boole (d. 1864) rigorous? That's a pretty bold claim.
    – charlotte
    Jan 24 '15 at 23:32






  • 2




    @charlotte: Cauchy is an interesting example. He tried to be rigorous, but he didn't quite get there! For instance, he "proved" that a pointwise limit of a sequence of continuous functions is continuous. On the other hand, his approach certainly wasn't "empirical" or "pragmatic" as Kline suggests.
    – user64687
    Jan 24 '15 at 23:36








  • 2




    I think "empirical" is a bit of sloppy use of language. Euler's product formula for the zeta function is an example. This was shown by Euler, but using techniques that did not have a rigorous foundation. But it wasn't merely observation - Euler made an argument, it just wasn't a complete argument.
    – Thomas Andrews
    Jan 24 '15 at 23:36














  • 7




    @abel: "Elements" is dated c. 300BC. Note that Kline says "reintroduced", not "introduced".
    – user64687
    Jan 24 '15 at 23:28






  • 6




    i think the word "rigour" as it relates proof is subjective. today some may say that some of of euler's proof are not rigorous.
    – abel
    Jan 24 '15 at 23:31








  • 6




    I guess he wouldn't consider Cauchy (d. 1857) or Boole (d. 1864) rigorous? That's a pretty bold claim.
    – charlotte
    Jan 24 '15 at 23:32






  • 2




    @charlotte: Cauchy is an interesting example. He tried to be rigorous, but he didn't quite get there! For instance, he "proved" that a pointwise limit of a sequence of continuous functions is continuous. On the other hand, his approach certainly wasn't "empirical" or "pragmatic" as Kline suggests.
    – user64687
    Jan 24 '15 at 23:36








  • 2




    I think "empirical" is a bit of sloppy use of language. Euler's product formula for the zeta function is an example. This was shown by Euler, but using techniques that did not have a rigorous foundation. But it wasn't merely observation - Euler made an argument, it just wasn't a complete argument.
    – Thomas Andrews
    Jan 24 '15 at 23:36








7




7




@abel: "Elements" is dated c. 300BC. Note that Kline says "reintroduced", not "introduced".
– user64687
Jan 24 '15 at 23:28




@abel: "Elements" is dated c. 300BC. Note that Kline says "reintroduced", not "introduced".
– user64687
Jan 24 '15 at 23:28




6




6




i think the word "rigour" as it relates proof is subjective. today some may say that some of of euler's proof are not rigorous.
– abel
Jan 24 '15 at 23:31






i think the word "rigour" as it relates proof is subjective. today some may say that some of of euler's proof are not rigorous.
– abel
Jan 24 '15 at 23:31






6




6




I guess he wouldn't consider Cauchy (d. 1857) or Boole (d. 1864) rigorous? That's a pretty bold claim.
– charlotte
Jan 24 '15 at 23:32




I guess he wouldn't consider Cauchy (d. 1857) or Boole (d. 1864) rigorous? That's a pretty bold claim.
– charlotte
Jan 24 '15 at 23:32




2




2




@charlotte: Cauchy is an interesting example. He tried to be rigorous, but he didn't quite get there! For instance, he "proved" that a pointwise limit of a sequence of continuous functions is continuous. On the other hand, his approach certainly wasn't "empirical" or "pragmatic" as Kline suggests.
– user64687
Jan 24 '15 at 23:36






@charlotte: Cauchy is an interesting example. He tried to be rigorous, but he didn't quite get there! For instance, he "proved" that a pointwise limit of a sequence of continuous functions is continuous. On the other hand, his approach certainly wasn't "empirical" or "pragmatic" as Kline suggests.
– user64687
Jan 24 '15 at 23:36






2




2




I think "empirical" is a bit of sloppy use of language. Euler's product formula for the zeta function is an example. This was shown by Euler, but using techniques that did not have a rigorous foundation. But it wasn't merely observation - Euler made an argument, it just wasn't a complete argument.
– Thomas Andrews
Jan 24 '15 at 23:36




I think "empirical" is a bit of sloppy use of language. Euler's product formula for the zeta function is an example. This was shown by Euler, but using techniques that did not have a rigorous foundation. But it wasn't merely observation - Euler made an argument, it just wasn't a complete argument.
– Thomas Andrews
Jan 24 '15 at 23:36










5 Answers
5






active

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10
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In mathematics, we prove results by deducing them from accepted truths that we call axioms. This is what Euclid taught us, and this is how mathematics is done until today. In my opinion, this wasn't substantially different between Euclid and Weierstrass, it's just that the mathematicians in the times of Fermat, Euler, Gauss or Dirichlet accepted more results as true than we do, because we have learned meanwhile how to prove some of these assumptions by reducing them to even fewer axioms.



To make clear what I mean by an example: Leibniz, Newton and Euler developed a technique for computing derivatives which they accepted as true and drew conclusions from there. Later, we learned how to prove that this technique worked by inventing the notion of limits. And then it became clear that certain assumptions about real numbers used in these proofs could be deduced from even more basic axioms.



As for Euclid: When Kline uses the Elements as a shining example for the axiomatic method he seems to forget that the Elements are not really more rigorous than the proofs of Euler; just look at Hilbert's foundations of geometry, where the holes in Euclid's proofs are addressed in detail.



Today we have a more or less accepted axiomatic system, but the situation is quite close to the "empirical and pragmatic" way of dpoing mathematics a few centuries ago: we believe that the axioms we are using today are consistent based on the fact that no contradictions have shown up for quite a long time. This is pragmatism based on empirical facts.






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  • Indeed I did - thanks!
    – franz lemmermeyer
    Nov 16 at 11:50


















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4
down vote













The proof of "Ptolemy's Theorem", in Almagest I, 10 is as rigorous as Euclid's Elements, which Ptolemy is clearly acquainted with and relying on, ca. 150 A.D. Even though Kline qualifies his claim by saying "almost all of mathematics..." it is still going too far to say "the concept of deductive proof from explicit axioms had been lost sight of."



Mathematicians may have run ahead of what they could prove, but they knew what proof was and should be. In his Two New Sciences, Third Day, offered as "discourses and demonstrations", Galileo's proof of Theorem I, the mean speed theorem for uniform acceleration, seems unrigorous in requiring us to view an area as composed of lines laid side by side--the "method of indivisibles" which later even Newton did not wholly rule out, provided it was used "judiciously". But it stands out as an exception. Galileo knows the standard, and, as far as he can, makes his dimostratzioni as rigorous as Euclid's.



On the other hand, even before 200 B.C., many of the theorems rigorously demonstrated by Archimedes, e.g. on centers of gravity, he first discovered and tested empirically.



Inquiry and discovery are one thing and proof is another, and this has never been lost sight of to the degree Kline seems to be claiming in the cited passage.






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    up vote
    4
    down vote













    If you read proofs by Euler or Gauss for example they don't use induction as we do today but talk about infinite descending - what is true for small numbers has to be true for big numbers. Maybe that gives you an impression what is meant by empirical proofs.



    Modern set theory and Zermelo-Fraenkel came up around 1900. But I would say that pure mathematics has taken place before - the areas just got renamed. In ancient Greece everything was Philosophy but fields like Algebra could make statements like the Pythagoras theorem which don't need much translation to modern theory.



    Mathematics still depends on intuition and subjectivity. Nobody would doubt that Ramanujan contributed a lot to modern mathematics even though his concept of proving was not compatible to the methods of mathematicians of his era.



    So in conclusion mathematics has only become canonical which means that proofs by god's input in dream are not accepted by the Annals of Mathematics.






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      +300










      Below I copy and paste wikipedia segments as well as a quote from Richard Dedekind. We concentrate here on just the nineteenth century, but see the last section where we pay tribute to Gottfried Wilhelm Leibniz.



      I don't agree with the author. Indeed, a couple of axioms needed to be pinned down, and set theory provided the glue, but logical and rigorous exposition of mathematical reasoning took a quantum leap during all of the nineteenth century.





      Augustin-Louis Cauchy:





      He was one of the first to state and rigorously prove theorems of calculus, rejecting the heuristic principle of the generality of algebra of earlier authors.





      Karl Weierstrass



      enter image description here



      Soundness of calculus



      Weierstrass was interested in the soundness of calculus, and at the time, there were somewhat ambiguous definitions regarding the foundations of calculus, and hence important theorems could not be proven with sufficient rigour. While Bolzano had developed a reasonably rigorous definition of a limit as early as 1817 (and possibly even earlier) his work remained unknown to most of the mathematical community until years later, and many mathematicians had only vague definitions of limits and continuity of functions.



      Delta-epsilon proofs are first found in the works of Cauchy in the 1820s. Cauchy did not clearly distinguish between continuity and uniform continuity on an interval. Notably, in his 1821 Cours d'analyse, Cauchy argued that the (pointwise) limit of (pointwise) continuous functions was itself (pointwise) continuous, a statement interpreted as being incorrect by many scholars. The correct statement is rather that the uniform limit of continuous functions is continuous (also, the uniform limit of uniformly continuous functions is uniformly continuous). This required the concept of uniform convergence, which was first observed by Weierstrass's advisor, Christoph Gudermann, in an 1838 paper, where Gudermann noted the phenomenon but did not define it or elaborate on it. Weierstrass saw the importance of the concept, and both formalized it and applied it widely throughout the foundations of calculus.





      Richard Dedekind



      enter image description here



      Julius Wilhelm Richard Dedekind (6 October 1831 – 12 February 1916) was a German mathematician who made important contributions to abstract algebra (particularly ring theory), axiomatic foundation for the natural numbers, algebraic number theory and the definition of the real numbers.




      That which is provable, ought not to be believed in science without
      proof.
      — Richard Dedekind (1888)






      REVOLUTION or CONSOLIDATION?



      Georg Cantor



      enter image description here



      Cantor established the importance of one-to-one correspondence between the members of two sets, defined infinite and well-ordered sets, and proved that the real numbers are more numerous than the natural numbers.





      Gottfried Wilhelm Leibniz:



      We find this quote from Benoit B. Mandelbrot discussing and quoting his hero, Leibniz:




      In Euclidis Prota ..., which is an attempt to tighten Euclid's axioms,
      he states ...: "I have diverse definitions for the straight line. The
      straight line is a curve, any part of which is similar to the whole,
      and it alone has this property, not only among curves but among sets."
      This claim can be proved today.




      So Leibniz was attempting to revamp the axioms found in Euclid's Elements.



      Sometimes you can pack a considerable amount of mathematics and logic into notation, but the Leibniz's notation is truly a remarkable achievement. Amazingly, wikipedia states




      In the modern rigorous treatment of non-standard calculus,
      justification can be found to again consider the notation as
      representing an actual quotient.




      I get the impression that Leibniz was the first person OF THIS WORLD interested in meta-mathematics; try a google search on keywords Gödel Leibniz.



      He is still in the news today:



      The Philosopher Who Helped Create the Information Age

      Gottfried Wilhelm Leibniz isn’t a household name—but he should be.



      Google doodle: Who was Gottfried Wilhelm Leibniz? What does the doodle mean?

      Today’s Google doodle celebrates the birth date of German polymath and philosopher Gottfried Wilhelm Leibniz. So who was he?






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      • 1




        And a long list of others! On the more physical side Newton and Leibniz are just two names that used a proof-like rigour.
        – R. Rankin
        2 days ago










      • @R.Rankin Didn't mean to slight Newton, but feel that it's a shame that Leibniz and Gödel were not contemporaries. see philosophy.stackexchange.com/questions/46956/…
        – CopyPasteIt
        2 days ago






      • 1




        @Copypastelt Oh I definitely agree. Also, we've got Fourier, Bernoulie brothers, Jean Le Rond D'Alembert. Actually there are soo many.
        – R. Rankin
        yesterday




















      up vote
      0
      down vote













      There is no distinction between applied math and pure math, especially during the discussed period. It was just math, with less rigour. Also, the idea that applied math has less rigour is an absurd one. This apparent division is not about rigour, it's about traditionally which branches have been called applied and which have been called pure.






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        In mathematics, we prove results by deducing them from accepted truths that we call axioms. This is what Euclid taught us, and this is how mathematics is done until today. In my opinion, this wasn't substantially different between Euclid and Weierstrass, it's just that the mathematicians in the times of Fermat, Euler, Gauss or Dirichlet accepted more results as true than we do, because we have learned meanwhile how to prove some of these assumptions by reducing them to even fewer axioms.



        To make clear what I mean by an example: Leibniz, Newton and Euler developed a technique for computing derivatives which they accepted as true and drew conclusions from there. Later, we learned how to prove that this technique worked by inventing the notion of limits. And then it became clear that certain assumptions about real numbers used in these proofs could be deduced from even more basic axioms.



        As for Euclid: When Kline uses the Elements as a shining example for the axiomatic method he seems to forget that the Elements are not really more rigorous than the proofs of Euler; just look at Hilbert's foundations of geometry, where the holes in Euclid's proofs are addressed in detail.



        Today we have a more or less accepted axiomatic system, but the situation is quite close to the "empirical and pragmatic" way of dpoing mathematics a few centuries ago: we believe that the axioms we are using today are consistent based on the fact that no contradictions have shown up for quite a long time. This is pragmatism based on empirical facts.






        share|cite|improve this answer























        • Indeed I did - thanks!
          – franz lemmermeyer
          Nov 16 at 11:50















        up vote
        10
        down vote













        In mathematics, we prove results by deducing them from accepted truths that we call axioms. This is what Euclid taught us, and this is how mathematics is done until today. In my opinion, this wasn't substantially different between Euclid and Weierstrass, it's just that the mathematicians in the times of Fermat, Euler, Gauss or Dirichlet accepted more results as true than we do, because we have learned meanwhile how to prove some of these assumptions by reducing them to even fewer axioms.



        To make clear what I mean by an example: Leibniz, Newton and Euler developed a technique for computing derivatives which they accepted as true and drew conclusions from there. Later, we learned how to prove that this technique worked by inventing the notion of limits. And then it became clear that certain assumptions about real numbers used in these proofs could be deduced from even more basic axioms.



        As for Euclid: When Kline uses the Elements as a shining example for the axiomatic method he seems to forget that the Elements are not really more rigorous than the proofs of Euler; just look at Hilbert's foundations of geometry, where the holes in Euclid's proofs are addressed in detail.



        Today we have a more or less accepted axiomatic system, but the situation is quite close to the "empirical and pragmatic" way of dpoing mathematics a few centuries ago: we believe that the axioms we are using today are consistent based on the fact that no contradictions have shown up for quite a long time. This is pragmatism based on empirical facts.






        share|cite|improve this answer























        • Indeed I did - thanks!
          – franz lemmermeyer
          Nov 16 at 11:50













        up vote
        10
        down vote










        up vote
        10
        down vote









        In mathematics, we prove results by deducing them from accepted truths that we call axioms. This is what Euclid taught us, and this is how mathematics is done until today. In my opinion, this wasn't substantially different between Euclid and Weierstrass, it's just that the mathematicians in the times of Fermat, Euler, Gauss or Dirichlet accepted more results as true than we do, because we have learned meanwhile how to prove some of these assumptions by reducing them to even fewer axioms.



        To make clear what I mean by an example: Leibniz, Newton and Euler developed a technique for computing derivatives which they accepted as true and drew conclusions from there. Later, we learned how to prove that this technique worked by inventing the notion of limits. And then it became clear that certain assumptions about real numbers used in these proofs could be deduced from even more basic axioms.



        As for Euclid: When Kline uses the Elements as a shining example for the axiomatic method he seems to forget that the Elements are not really more rigorous than the proofs of Euler; just look at Hilbert's foundations of geometry, where the holes in Euclid's proofs are addressed in detail.



        Today we have a more or less accepted axiomatic system, but the situation is quite close to the "empirical and pragmatic" way of dpoing mathematics a few centuries ago: we believe that the axioms we are using today are consistent based on the fact that no contradictions have shown up for quite a long time. This is pragmatism based on empirical facts.






        share|cite|improve this answer














        In mathematics, we prove results by deducing them from accepted truths that we call axioms. This is what Euclid taught us, and this is how mathematics is done until today. In my opinion, this wasn't substantially different between Euclid and Weierstrass, it's just that the mathematicians in the times of Fermat, Euler, Gauss or Dirichlet accepted more results as true than we do, because we have learned meanwhile how to prove some of these assumptions by reducing them to even fewer axioms.



        To make clear what I mean by an example: Leibniz, Newton and Euler developed a technique for computing derivatives which they accepted as true and drew conclusions from there. Later, we learned how to prove that this technique worked by inventing the notion of limits. And then it became clear that certain assumptions about real numbers used in these proofs could be deduced from even more basic axioms.



        As for Euclid: When Kline uses the Elements as a shining example for the axiomatic method he seems to forget that the Elements are not really more rigorous than the proofs of Euler; just look at Hilbert's foundations of geometry, where the holes in Euclid's proofs are addressed in detail.



        Today we have a more or less accepted axiomatic system, but the situation is quite close to the "empirical and pragmatic" way of dpoing mathematics a few centuries ago: we believe that the axioms we are using today are consistent based on the fact that no contradictions have shown up for quite a long time. This is pragmatism based on empirical facts.







        share|cite|improve this answer














        share|cite|improve this answer



        share|cite|improve this answer








        edited Nov 16 at 11:50

























        answered Nov 15 at 17:37









        franz lemmermeyer

        6,86521847




        6,86521847












        • Indeed I did - thanks!
          – franz lemmermeyer
          Nov 16 at 11:50


















        • Indeed I did - thanks!
          – franz lemmermeyer
          Nov 16 at 11:50
















        Indeed I did - thanks!
        – franz lemmermeyer
        Nov 16 at 11:50




        Indeed I did - thanks!
        – franz lemmermeyer
        Nov 16 at 11:50










        up vote
        4
        down vote













        The proof of "Ptolemy's Theorem", in Almagest I, 10 is as rigorous as Euclid's Elements, which Ptolemy is clearly acquainted with and relying on, ca. 150 A.D. Even though Kline qualifies his claim by saying "almost all of mathematics..." it is still going too far to say "the concept of deductive proof from explicit axioms had been lost sight of."



        Mathematicians may have run ahead of what they could prove, but they knew what proof was and should be. In his Two New Sciences, Third Day, offered as "discourses and demonstrations", Galileo's proof of Theorem I, the mean speed theorem for uniform acceleration, seems unrigorous in requiring us to view an area as composed of lines laid side by side--the "method of indivisibles" which later even Newton did not wholly rule out, provided it was used "judiciously". But it stands out as an exception. Galileo knows the standard, and, as far as he can, makes his dimostratzioni as rigorous as Euclid's.



        On the other hand, even before 200 B.C., many of the theorems rigorously demonstrated by Archimedes, e.g. on centers of gravity, he first discovered and tested empirically.



        Inquiry and discovery are one thing and proof is another, and this has never been lost sight of to the degree Kline seems to be claiming in the cited passage.






        share|cite|improve this answer

























          up vote
          4
          down vote













          The proof of "Ptolemy's Theorem", in Almagest I, 10 is as rigorous as Euclid's Elements, which Ptolemy is clearly acquainted with and relying on, ca. 150 A.D. Even though Kline qualifies his claim by saying "almost all of mathematics..." it is still going too far to say "the concept of deductive proof from explicit axioms had been lost sight of."



          Mathematicians may have run ahead of what they could prove, but they knew what proof was and should be. In his Two New Sciences, Third Day, offered as "discourses and demonstrations", Galileo's proof of Theorem I, the mean speed theorem for uniform acceleration, seems unrigorous in requiring us to view an area as composed of lines laid side by side--the "method of indivisibles" which later even Newton did not wholly rule out, provided it was used "judiciously". But it stands out as an exception. Galileo knows the standard, and, as far as he can, makes his dimostratzioni as rigorous as Euclid's.



          On the other hand, even before 200 B.C., many of the theorems rigorously demonstrated by Archimedes, e.g. on centers of gravity, he first discovered and tested empirically.



          Inquiry and discovery are one thing and proof is another, and this has never been lost sight of to the degree Kline seems to be claiming in the cited passage.






          share|cite|improve this answer























            up vote
            4
            down vote










            up vote
            4
            down vote









            The proof of "Ptolemy's Theorem", in Almagest I, 10 is as rigorous as Euclid's Elements, which Ptolemy is clearly acquainted with and relying on, ca. 150 A.D. Even though Kline qualifies his claim by saying "almost all of mathematics..." it is still going too far to say "the concept of deductive proof from explicit axioms had been lost sight of."



            Mathematicians may have run ahead of what they could prove, but they knew what proof was and should be. In his Two New Sciences, Third Day, offered as "discourses and demonstrations", Galileo's proof of Theorem I, the mean speed theorem for uniform acceleration, seems unrigorous in requiring us to view an area as composed of lines laid side by side--the "method of indivisibles" which later even Newton did not wholly rule out, provided it was used "judiciously". But it stands out as an exception. Galileo knows the standard, and, as far as he can, makes his dimostratzioni as rigorous as Euclid's.



            On the other hand, even before 200 B.C., many of the theorems rigorously demonstrated by Archimedes, e.g. on centers of gravity, he first discovered and tested empirically.



            Inquiry and discovery are one thing and proof is another, and this has never been lost sight of to the degree Kline seems to be claiming in the cited passage.






            share|cite|improve this answer












            The proof of "Ptolemy's Theorem", in Almagest I, 10 is as rigorous as Euclid's Elements, which Ptolemy is clearly acquainted with and relying on, ca. 150 A.D. Even though Kline qualifies his claim by saying "almost all of mathematics..." it is still going too far to say "the concept of deductive proof from explicit axioms had been lost sight of."



            Mathematicians may have run ahead of what they could prove, but they knew what proof was and should be. In his Two New Sciences, Third Day, offered as "discourses and demonstrations", Galileo's proof of Theorem I, the mean speed theorem for uniform acceleration, seems unrigorous in requiring us to view an area as composed of lines laid side by side--the "method of indivisibles" which later even Newton did not wholly rule out, provided it was used "judiciously". But it stands out as an exception. Galileo knows the standard, and, as far as he can, makes his dimostratzioni as rigorous as Euclid's.



            On the other hand, even before 200 B.C., many of the theorems rigorously demonstrated by Archimedes, e.g. on centers of gravity, he first discovered and tested empirically.



            Inquiry and discovery are one thing and proof is another, and this has never been lost sight of to the degree Kline seems to be claiming in the cited passage.







            share|cite|improve this answer












            share|cite|improve this answer



            share|cite|improve this answer










            answered Nov 15 at 20:13









            Edward Porcella

            1,3661411




            1,3661411






















                up vote
                4
                down vote













                If you read proofs by Euler or Gauss for example they don't use induction as we do today but talk about infinite descending - what is true for small numbers has to be true for big numbers. Maybe that gives you an impression what is meant by empirical proofs.



                Modern set theory and Zermelo-Fraenkel came up around 1900. But I would say that pure mathematics has taken place before - the areas just got renamed. In ancient Greece everything was Philosophy but fields like Algebra could make statements like the Pythagoras theorem which don't need much translation to modern theory.



                Mathematics still depends on intuition and subjectivity. Nobody would doubt that Ramanujan contributed a lot to modern mathematics even though his concept of proving was not compatible to the methods of mathematicians of his era.



                So in conclusion mathematics has only become canonical which means that proofs by god's input in dream are not accepted by the Annals of Mathematics.






                share|cite|improve this answer








                New contributor




                Nodt Greenish is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
                Check out our Code of Conduct.






















                  up vote
                  4
                  down vote













                  If you read proofs by Euler or Gauss for example they don't use induction as we do today but talk about infinite descending - what is true for small numbers has to be true for big numbers. Maybe that gives you an impression what is meant by empirical proofs.



                  Modern set theory and Zermelo-Fraenkel came up around 1900. But I would say that pure mathematics has taken place before - the areas just got renamed. In ancient Greece everything was Philosophy but fields like Algebra could make statements like the Pythagoras theorem which don't need much translation to modern theory.



                  Mathematics still depends on intuition and subjectivity. Nobody would doubt that Ramanujan contributed a lot to modern mathematics even though his concept of proving was not compatible to the methods of mathematicians of his era.



                  So in conclusion mathematics has only become canonical which means that proofs by god's input in dream are not accepted by the Annals of Mathematics.






                  share|cite|improve this answer








                  New contributor




                  Nodt Greenish is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
                  Check out our Code of Conduct.




















                    up vote
                    4
                    down vote










                    up vote
                    4
                    down vote









                    If you read proofs by Euler or Gauss for example they don't use induction as we do today but talk about infinite descending - what is true for small numbers has to be true for big numbers. Maybe that gives you an impression what is meant by empirical proofs.



                    Modern set theory and Zermelo-Fraenkel came up around 1900. But I would say that pure mathematics has taken place before - the areas just got renamed. In ancient Greece everything was Philosophy but fields like Algebra could make statements like the Pythagoras theorem which don't need much translation to modern theory.



                    Mathematics still depends on intuition and subjectivity. Nobody would doubt that Ramanujan contributed a lot to modern mathematics even though his concept of proving was not compatible to the methods of mathematicians of his era.



                    So in conclusion mathematics has only become canonical which means that proofs by god's input in dream are not accepted by the Annals of Mathematics.






                    share|cite|improve this answer








                    New contributor




                    Nodt Greenish is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
                    Check out our Code of Conduct.









                    If you read proofs by Euler or Gauss for example they don't use induction as we do today but talk about infinite descending - what is true for small numbers has to be true for big numbers. Maybe that gives you an impression what is meant by empirical proofs.



                    Modern set theory and Zermelo-Fraenkel came up around 1900. But I would say that pure mathematics has taken place before - the areas just got renamed. In ancient Greece everything was Philosophy but fields like Algebra could make statements like the Pythagoras theorem which don't need much translation to modern theory.



                    Mathematics still depends on intuition and subjectivity. Nobody would doubt that Ramanujan contributed a lot to modern mathematics even though his concept of proving was not compatible to the methods of mathematicians of his era.



                    So in conclusion mathematics has only become canonical which means that proofs by god's input in dream are not accepted by the Annals of Mathematics.







                    share|cite|improve this answer








                    New contributor




                    Nodt Greenish is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
                    Check out our Code of Conduct.









                    share|cite|improve this answer



                    share|cite|improve this answer






                    New contributor




                    Nodt Greenish is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
                    Check out our Code of Conduct.









                    answered Nov 17 at 10:26









                    Nodt Greenish

                    12010




                    12010




                    New contributor




                    Nodt Greenish is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
                    Check out our Code of Conduct.





                    New contributor





                    Nodt Greenish is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
                    Check out our Code of Conduct.






                    Nodt Greenish is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
                    Check out our Code of Conduct.






















                        up vote
                        4
                        down vote



                        +300










                        Below I copy and paste wikipedia segments as well as a quote from Richard Dedekind. We concentrate here on just the nineteenth century, but see the last section where we pay tribute to Gottfried Wilhelm Leibniz.



                        I don't agree with the author. Indeed, a couple of axioms needed to be pinned down, and set theory provided the glue, but logical and rigorous exposition of mathematical reasoning took a quantum leap during all of the nineteenth century.





                        Augustin-Louis Cauchy:





                        He was one of the first to state and rigorously prove theorems of calculus, rejecting the heuristic principle of the generality of algebra of earlier authors.





                        Karl Weierstrass



                        enter image description here



                        Soundness of calculus



                        Weierstrass was interested in the soundness of calculus, and at the time, there were somewhat ambiguous definitions regarding the foundations of calculus, and hence important theorems could not be proven with sufficient rigour. While Bolzano had developed a reasonably rigorous definition of a limit as early as 1817 (and possibly even earlier) his work remained unknown to most of the mathematical community until years later, and many mathematicians had only vague definitions of limits and continuity of functions.



                        Delta-epsilon proofs are first found in the works of Cauchy in the 1820s. Cauchy did not clearly distinguish between continuity and uniform continuity on an interval. Notably, in his 1821 Cours d'analyse, Cauchy argued that the (pointwise) limit of (pointwise) continuous functions was itself (pointwise) continuous, a statement interpreted as being incorrect by many scholars. The correct statement is rather that the uniform limit of continuous functions is continuous (also, the uniform limit of uniformly continuous functions is uniformly continuous). This required the concept of uniform convergence, which was first observed by Weierstrass's advisor, Christoph Gudermann, in an 1838 paper, where Gudermann noted the phenomenon but did not define it or elaborate on it. Weierstrass saw the importance of the concept, and both formalized it and applied it widely throughout the foundations of calculus.





                        Richard Dedekind



                        enter image description here



                        Julius Wilhelm Richard Dedekind (6 October 1831 – 12 February 1916) was a German mathematician who made important contributions to abstract algebra (particularly ring theory), axiomatic foundation for the natural numbers, algebraic number theory and the definition of the real numbers.




                        That which is provable, ought not to be believed in science without
                        proof.
                        — Richard Dedekind (1888)






                        REVOLUTION or CONSOLIDATION?



                        Georg Cantor



                        enter image description here



                        Cantor established the importance of one-to-one correspondence between the members of two sets, defined infinite and well-ordered sets, and proved that the real numbers are more numerous than the natural numbers.





                        Gottfried Wilhelm Leibniz:



                        We find this quote from Benoit B. Mandelbrot discussing and quoting his hero, Leibniz:




                        In Euclidis Prota ..., which is an attempt to tighten Euclid's axioms,
                        he states ...: "I have diverse definitions for the straight line. The
                        straight line is a curve, any part of which is similar to the whole,
                        and it alone has this property, not only among curves but among sets."
                        This claim can be proved today.




                        So Leibniz was attempting to revamp the axioms found in Euclid's Elements.



                        Sometimes you can pack a considerable amount of mathematics and logic into notation, but the Leibniz's notation is truly a remarkable achievement. Amazingly, wikipedia states




                        In the modern rigorous treatment of non-standard calculus,
                        justification can be found to again consider the notation as
                        representing an actual quotient.




                        I get the impression that Leibniz was the first person OF THIS WORLD interested in meta-mathematics; try a google search on keywords Gödel Leibniz.



                        He is still in the news today:



                        The Philosopher Who Helped Create the Information Age

                        Gottfried Wilhelm Leibniz isn’t a household name—but he should be.



                        Google doodle: Who was Gottfried Wilhelm Leibniz? What does the doodle mean?

                        Today’s Google doodle celebrates the birth date of German polymath and philosopher Gottfried Wilhelm Leibniz. So who was he?






                        share|cite|improve this answer



















                        • 1




                          And a long list of others! On the more physical side Newton and Leibniz are just two names that used a proof-like rigour.
                          – R. Rankin
                          2 days ago










                        • @R.Rankin Didn't mean to slight Newton, but feel that it's a shame that Leibniz and Gödel were not contemporaries. see philosophy.stackexchange.com/questions/46956/…
                          – CopyPasteIt
                          2 days ago






                        • 1




                          @Copypastelt Oh I definitely agree. Also, we've got Fourier, Bernoulie brothers, Jean Le Rond D'Alembert. Actually there are soo many.
                          – R. Rankin
                          yesterday

















                        up vote
                        4
                        down vote



                        +300










                        Below I copy and paste wikipedia segments as well as a quote from Richard Dedekind. We concentrate here on just the nineteenth century, but see the last section where we pay tribute to Gottfried Wilhelm Leibniz.



                        I don't agree with the author. Indeed, a couple of axioms needed to be pinned down, and set theory provided the glue, but logical and rigorous exposition of mathematical reasoning took a quantum leap during all of the nineteenth century.





                        Augustin-Louis Cauchy:





                        He was one of the first to state and rigorously prove theorems of calculus, rejecting the heuristic principle of the generality of algebra of earlier authors.





                        Karl Weierstrass



                        enter image description here



                        Soundness of calculus



                        Weierstrass was interested in the soundness of calculus, and at the time, there were somewhat ambiguous definitions regarding the foundations of calculus, and hence important theorems could not be proven with sufficient rigour. While Bolzano had developed a reasonably rigorous definition of a limit as early as 1817 (and possibly even earlier) his work remained unknown to most of the mathematical community until years later, and many mathematicians had only vague definitions of limits and continuity of functions.



                        Delta-epsilon proofs are first found in the works of Cauchy in the 1820s. Cauchy did not clearly distinguish between continuity and uniform continuity on an interval. Notably, in his 1821 Cours d'analyse, Cauchy argued that the (pointwise) limit of (pointwise) continuous functions was itself (pointwise) continuous, a statement interpreted as being incorrect by many scholars. The correct statement is rather that the uniform limit of continuous functions is continuous (also, the uniform limit of uniformly continuous functions is uniformly continuous). This required the concept of uniform convergence, which was first observed by Weierstrass's advisor, Christoph Gudermann, in an 1838 paper, where Gudermann noted the phenomenon but did not define it or elaborate on it. Weierstrass saw the importance of the concept, and both formalized it and applied it widely throughout the foundations of calculus.





                        Richard Dedekind



                        enter image description here



                        Julius Wilhelm Richard Dedekind (6 October 1831 – 12 February 1916) was a German mathematician who made important contributions to abstract algebra (particularly ring theory), axiomatic foundation for the natural numbers, algebraic number theory and the definition of the real numbers.




                        That which is provable, ought not to be believed in science without
                        proof.
                        — Richard Dedekind (1888)






                        REVOLUTION or CONSOLIDATION?



                        Georg Cantor



                        enter image description here



                        Cantor established the importance of one-to-one correspondence between the members of two sets, defined infinite and well-ordered sets, and proved that the real numbers are more numerous than the natural numbers.





                        Gottfried Wilhelm Leibniz:



                        We find this quote from Benoit B. Mandelbrot discussing and quoting his hero, Leibniz:




                        In Euclidis Prota ..., which is an attempt to tighten Euclid's axioms,
                        he states ...: "I have diverse definitions for the straight line. The
                        straight line is a curve, any part of which is similar to the whole,
                        and it alone has this property, not only among curves but among sets."
                        This claim can be proved today.




                        So Leibniz was attempting to revamp the axioms found in Euclid's Elements.



                        Sometimes you can pack a considerable amount of mathematics and logic into notation, but the Leibniz's notation is truly a remarkable achievement. Amazingly, wikipedia states




                        In the modern rigorous treatment of non-standard calculus,
                        justification can be found to again consider the notation as
                        representing an actual quotient.




                        I get the impression that Leibniz was the first person OF THIS WORLD interested in meta-mathematics; try a google search on keywords Gödel Leibniz.



                        He is still in the news today:



                        The Philosopher Who Helped Create the Information Age

                        Gottfried Wilhelm Leibniz isn’t a household name—but he should be.



                        Google doodle: Who was Gottfried Wilhelm Leibniz? What does the doodle mean?

                        Today’s Google doodle celebrates the birth date of German polymath and philosopher Gottfried Wilhelm Leibniz. So who was he?






                        share|cite|improve this answer



















                        • 1




                          And a long list of others! On the more physical side Newton and Leibniz are just two names that used a proof-like rigour.
                          – R. Rankin
                          2 days ago










                        • @R.Rankin Didn't mean to slight Newton, but feel that it's a shame that Leibniz and Gödel were not contemporaries. see philosophy.stackexchange.com/questions/46956/…
                          – CopyPasteIt
                          2 days ago






                        • 1




                          @Copypastelt Oh I definitely agree. Also, we've got Fourier, Bernoulie brothers, Jean Le Rond D'Alembert. Actually there are soo many.
                          – R. Rankin
                          yesterday















                        up vote
                        4
                        down vote



                        +300







                        up vote
                        4
                        down vote



                        +300




                        +300




                        Below I copy and paste wikipedia segments as well as a quote from Richard Dedekind. We concentrate here on just the nineteenth century, but see the last section where we pay tribute to Gottfried Wilhelm Leibniz.



                        I don't agree with the author. Indeed, a couple of axioms needed to be pinned down, and set theory provided the glue, but logical and rigorous exposition of mathematical reasoning took a quantum leap during all of the nineteenth century.





                        Augustin-Louis Cauchy:





                        He was one of the first to state and rigorously prove theorems of calculus, rejecting the heuristic principle of the generality of algebra of earlier authors.





                        Karl Weierstrass



                        enter image description here



                        Soundness of calculus



                        Weierstrass was interested in the soundness of calculus, and at the time, there were somewhat ambiguous definitions regarding the foundations of calculus, and hence important theorems could not be proven with sufficient rigour. While Bolzano had developed a reasonably rigorous definition of a limit as early as 1817 (and possibly even earlier) his work remained unknown to most of the mathematical community until years later, and many mathematicians had only vague definitions of limits and continuity of functions.



                        Delta-epsilon proofs are first found in the works of Cauchy in the 1820s. Cauchy did not clearly distinguish between continuity and uniform continuity on an interval. Notably, in his 1821 Cours d'analyse, Cauchy argued that the (pointwise) limit of (pointwise) continuous functions was itself (pointwise) continuous, a statement interpreted as being incorrect by many scholars. The correct statement is rather that the uniform limit of continuous functions is continuous (also, the uniform limit of uniformly continuous functions is uniformly continuous). This required the concept of uniform convergence, which was first observed by Weierstrass's advisor, Christoph Gudermann, in an 1838 paper, where Gudermann noted the phenomenon but did not define it or elaborate on it. Weierstrass saw the importance of the concept, and both formalized it and applied it widely throughout the foundations of calculus.





                        Richard Dedekind



                        enter image description here



                        Julius Wilhelm Richard Dedekind (6 October 1831 – 12 February 1916) was a German mathematician who made important contributions to abstract algebra (particularly ring theory), axiomatic foundation for the natural numbers, algebraic number theory and the definition of the real numbers.




                        That which is provable, ought not to be believed in science without
                        proof.
                        — Richard Dedekind (1888)






                        REVOLUTION or CONSOLIDATION?



                        Georg Cantor



                        enter image description here



                        Cantor established the importance of one-to-one correspondence between the members of two sets, defined infinite and well-ordered sets, and proved that the real numbers are more numerous than the natural numbers.





                        Gottfried Wilhelm Leibniz:



                        We find this quote from Benoit B. Mandelbrot discussing and quoting his hero, Leibniz:




                        In Euclidis Prota ..., which is an attempt to tighten Euclid's axioms,
                        he states ...: "I have diverse definitions for the straight line. The
                        straight line is a curve, any part of which is similar to the whole,
                        and it alone has this property, not only among curves but among sets."
                        This claim can be proved today.




                        So Leibniz was attempting to revamp the axioms found in Euclid's Elements.



                        Sometimes you can pack a considerable amount of mathematics and logic into notation, but the Leibniz's notation is truly a remarkable achievement. Amazingly, wikipedia states




                        In the modern rigorous treatment of non-standard calculus,
                        justification can be found to again consider the notation as
                        representing an actual quotient.




                        I get the impression that Leibniz was the first person OF THIS WORLD interested in meta-mathematics; try a google search on keywords Gödel Leibniz.



                        He is still in the news today:



                        The Philosopher Who Helped Create the Information Age

                        Gottfried Wilhelm Leibniz isn’t a household name—but he should be.



                        Google doodle: Who was Gottfried Wilhelm Leibniz? What does the doodle mean?

                        Today’s Google doodle celebrates the birth date of German polymath and philosopher Gottfried Wilhelm Leibniz. So who was he?






                        share|cite|improve this answer














                        Below I copy and paste wikipedia segments as well as a quote from Richard Dedekind. We concentrate here on just the nineteenth century, but see the last section where we pay tribute to Gottfried Wilhelm Leibniz.



                        I don't agree with the author. Indeed, a couple of axioms needed to be pinned down, and set theory provided the glue, but logical and rigorous exposition of mathematical reasoning took a quantum leap during all of the nineteenth century.





                        Augustin-Louis Cauchy:





                        He was one of the first to state and rigorously prove theorems of calculus, rejecting the heuristic principle of the generality of algebra of earlier authors.





                        Karl Weierstrass



                        enter image description here



                        Soundness of calculus



                        Weierstrass was interested in the soundness of calculus, and at the time, there were somewhat ambiguous definitions regarding the foundations of calculus, and hence important theorems could not be proven with sufficient rigour. While Bolzano had developed a reasonably rigorous definition of a limit as early as 1817 (and possibly even earlier) his work remained unknown to most of the mathematical community until years later, and many mathematicians had only vague definitions of limits and continuity of functions.



                        Delta-epsilon proofs are first found in the works of Cauchy in the 1820s. Cauchy did not clearly distinguish between continuity and uniform continuity on an interval. Notably, in his 1821 Cours d'analyse, Cauchy argued that the (pointwise) limit of (pointwise) continuous functions was itself (pointwise) continuous, a statement interpreted as being incorrect by many scholars. The correct statement is rather that the uniform limit of continuous functions is continuous (also, the uniform limit of uniformly continuous functions is uniformly continuous). This required the concept of uniform convergence, which was first observed by Weierstrass's advisor, Christoph Gudermann, in an 1838 paper, where Gudermann noted the phenomenon but did not define it or elaborate on it. Weierstrass saw the importance of the concept, and both formalized it and applied it widely throughout the foundations of calculus.





                        Richard Dedekind



                        enter image description here



                        Julius Wilhelm Richard Dedekind (6 October 1831 – 12 February 1916) was a German mathematician who made important contributions to abstract algebra (particularly ring theory), axiomatic foundation for the natural numbers, algebraic number theory and the definition of the real numbers.




                        That which is provable, ought not to be believed in science without
                        proof.
                        — Richard Dedekind (1888)






                        REVOLUTION or CONSOLIDATION?



                        Georg Cantor



                        enter image description here



                        Cantor established the importance of one-to-one correspondence between the members of two sets, defined infinite and well-ordered sets, and proved that the real numbers are more numerous than the natural numbers.





                        Gottfried Wilhelm Leibniz:



                        We find this quote from Benoit B. Mandelbrot discussing and quoting his hero, Leibniz:




                        In Euclidis Prota ..., which is an attempt to tighten Euclid's axioms,
                        he states ...: "I have diverse definitions for the straight line. The
                        straight line is a curve, any part of which is similar to the whole,
                        and it alone has this property, not only among curves but among sets."
                        This claim can be proved today.




                        So Leibniz was attempting to revamp the axioms found in Euclid's Elements.



                        Sometimes you can pack a considerable amount of mathematics and logic into notation, but the Leibniz's notation is truly a remarkable achievement. Amazingly, wikipedia states




                        In the modern rigorous treatment of non-standard calculus,
                        justification can be found to again consider the notation as
                        representing an actual quotient.




                        I get the impression that Leibniz was the first person OF THIS WORLD interested in meta-mathematics; try a google search on keywords Gödel Leibniz.



                        He is still in the news today:



                        The Philosopher Who Helped Create the Information Age

                        Gottfried Wilhelm Leibniz isn’t a household name—but he should be.



                        Google doodle: Who was Gottfried Wilhelm Leibniz? What does the doodle mean?

                        Today’s Google doodle celebrates the birth date of German polymath and philosopher Gottfried Wilhelm Leibniz. So who was he?







                        share|cite|improve this answer














                        share|cite|improve this answer



                        share|cite|improve this answer








                        edited 2 days ago

























                        answered Nov 20 at 3:21









                        CopyPasteIt

                        3,7221527




                        3,7221527








                        • 1




                          And a long list of others! On the more physical side Newton and Leibniz are just two names that used a proof-like rigour.
                          – R. Rankin
                          2 days ago










                        • @R.Rankin Didn't mean to slight Newton, but feel that it's a shame that Leibniz and Gödel were not contemporaries. see philosophy.stackexchange.com/questions/46956/…
                          – CopyPasteIt
                          2 days ago






                        • 1




                          @Copypastelt Oh I definitely agree. Also, we've got Fourier, Bernoulie brothers, Jean Le Rond D'Alembert. Actually there are soo many.
                          – R. Rankin
                          yesterday
















                        • 1




                          And a long list of others! On the more physical side Newton and Leibniz are just two names that used a proof-like rigour.
                          – R. Rankin
                          2 days ago










                        • @R.Rankin Didn't mean to slight Newton, but feel that it's a shame that Leibniz and Gödel were not contemporaries. see philosophy.stackexchange.com/questions/46956/…
                          – CopyPasteIt
                          2 days ago






                        • 1




                          @Copypastelt Oh I definitely agree. Also, we've got Fourier, Bernoulie brothers, Jean Le Rond D'Alembert. Actually there are soo many.
                          – R. Rankin
                          yesterday










                        1




                        1




                        And a long list of others! On the more physical side Newton and Leibniz are just two names that used a proof-like rigour.
                        – R. Rankin
                        2 days ago




                        And a long list of others! On the more physical side Newton and Leibniz are just two names that used a proof-like rigour.
                        – R. Rankin
                        2 days ago












                        @R.Rankin Didn't mean to slight Newton, but feel that it's a shame that Leibniz and Gödel were not contemporaries. see philosophy.stackexchange.com/questions/46956/…
                        – CopyPasteIt
                        2 days ago




                        @R.Rankin Didn't mean to slight Newton, but feel that it's a shame that Leibniz and Gödel were not contemporaries. see philosophy.stackexchange.com/questions/46956/…
                        – CopyPasteIt
                        2 days ago




                        1




                        1




                        @Copypastelt Oh I definitely agree. Also, we've got Fourier, Bernoulie brothers, Jean Le Rond D'Alembert. Actually there are soo many.
                        – R. Rankin
                        yesterday






                        @Copypastelt Oh I definitely agree. Also, we've got Fourier, Bernoulie brothers, Jean Le Rond D'Alembert. Actually there are soo many.
                        – R. Rankin
                        yesterday












                        up vote
                        0
                        down vote













                        There is no distinction between applied math and pure math, especially during the discussed period. It was just math, with less rigour. Also, the idea that applied math has less rigour is an absurd one. This apparent division is not about rigour, it's about traditionally which branches have been called applied and which have been called pure.






                        share|cite|improve this answer

























                          up vote
                          0
                          down vote













                          There is no distinction between applied math and pure math, especially during the discussed period. It was just math, with less rigour. Also, the idea that applied math has less rigour is an absurd one. This apparent division is not about rigour, it's about traditionally which branches have been called applied and which have been called pure.






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                            up vote
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                            There is no distinction between applied math and pure math, especially during the discussed period. It was just math, with less rigour. Also, the idea that applied math has less rigour is an absurd one. This apparent division is not about rigour, it's about traditionally which branches have been called applied and which have been called pure.






                            share|cite|improve this answer












                            There is no distinction between applied math and pure math, especially during the discussed period. It was just math, with less rigour. Also, the idea that applied math has less rigour is an absurd one. This apparent division is not about rigour, it's about traditionally which branches have been called applied and which have been called pure.







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                            answered 2 days ago









                            timur

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                            11.5k1943






























                                 

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