مبانی «برنهاد منطق ریاضی» راسل

نوع مقاله : مقاله علمی- پژوهشی

نویسندگان

1 کارشناسی ارشد فلسفه، دانشگاه تبریز

2 دانشیار دانشگاه تبریز

چکیده

با در نظر گرفتن دو جریان فراگیر فلسفی قرن نوزدهم یعنی ایده‌آلیسم و رئالیسم، آنچه در این مقاله مورد بررسی قرار می­گیرد این است که راسل بر پایه­ کدام نظام فلسفی منطق­گرایی خود را بنا کرد. طرح منطق­گرایی راسل در سه دوره­ فکری بررسی می­شود. در دوره­ اوّل راسل در بررسی­های منطقی خویش رویکرد ایده‌آلیستی دارد. در دوره­ دوّم با پایه­ریزی فلسفه­ رئالیستی خود نسخه­ اوّلیه­ منطق­گرایی­اش را در اصول می­آورد. در دوره­ سوّم به دلیل پارادوکس­های برخواسته از جریان منطق­گرایی، راسل به ایده‌آلیسم اصلاح یافته­ای روی می­آورد و با ارائه­ «نظریه طبقات» نسخه­ کاملی از منطق­گرایی خود را در کتاب مبانیریاضی ارائه می­دهد. امّا راسل همچنان در ارائه­ عمومیتی معتبر که برای منطق و ریاضیات اساسی است با موانعی مواجه می­شود. نهایتا راسل دو «اصل موضوع تحویل­پذیری» و «اصل موضوع بی­نهایت» را برای برداشتن این موانع مطرح می­کند تا بتواند نظام منطقی خود را به درستی بنا کند.

تازه های تحقیق

Extended Abstract

Foundations of Russell’s “Logicist Thesis”

Masoomeh Ali Hassan Zadeh1 Masoud Omid2,

1Graduate of philosophy, Tabriz University

masoomehasanzade@gmail.com

2Associated Professor of philosophy department, Tabriz University

masoud_omid1345@yahoo.com

Abstract

This paper is concerned with a general account of logicism as developed within Russell’s philosophy of mathematics. At first Russell’s approach in metaphysic was idealistic and he relied on this point of view for extending his platonic atomism. To expound this account it will be demonstrated that after developing platonic atomism, Russell attempted to present his logicism in The Principles of Mathematics as a view opposed to an idealistic account of mathematics. However, a number of paradoxes arose had their roots deep in Russell’s metaphysical views. At this point he was forced to moderate his realistic extreme approach. Afterward it is shown that to evade these paradoxes, Russell adopts a view that allows for ontological distinctions and then introduces a full-fledged theory of types in Principia Mathematica. Nevertheless, the new framework yields problems of its own that pose a threat to Russell’s object-centered metaphysics but also deprives him of handling truths of unrestricted generality. Then to present a final version of his logicism, Russell’s way out of these issues will be set forth which comes in form of axioms of reducibility and infinity. Although these axioms represent a solution to remove paradoxes, actually those are too complicated to use.

Keywords: Logicism, Philosophy of Mathematics, Set, Foundation of mathematics, Proposition

 

  1. 1.      Introduction 

The logicists’ point of view is that mathematics is a branch of logic and logic precedes mathematics rather than being a mathematical tool. In this point of view all the mathematical concepts should be codify in terms of logical concepts and all the mathematical propositions must be extend as the propositions of logic. They believe logic is at the base and the distinction between logic and mathematics isn’t real (Eves, Howard Whitley: 321). We aim to explain Russell’s reduction of mathematics to logic. Russell’s logicism is concernd with his metaphysics, On the other hand he extended his logicism based on his metaphysics. We can say all the problem in this project raise from this connection. At first Russell adupt idealist views under the influence of Cambridge atmosphere (Hylton: 72). The idealists believe an Organic Unity, which amounts to more than the conjuction of the knowledge of one part with the knowledge of another part, and so on for each part. Actually they believe the idea of degrees of truth. But for Russell as a logician it was unavoidable to believe that there are no degrees of reality, either a proposition is true or it is false, the world can be known by seprate, distinct object.this was his Platonic Atomism that was opposite to idealistic views. On the other hand idealism says that all our ordinary (non-metaphysical) ways of thinking about the world are contradictory or incoherence.He held that the logic which he drived in part from Peano was capable of showing the whole of mathematics, including geometry, to be perfectly coherent and consistent. The method by which this was to be achieved was the reduction of mathematics to logic i,e, the demonstration that mathematical concepts can be understood as built of from the basic concepts of logic, and that mathematical truths can all be proved from the basic truths of logic. This demonstration was outlined and discussed in Principles (Hylton: 112-115) . However, a number of paradoxes arose that had their roots deep in Russell’s metaphysical views. Russell claims that a single error lies behind all of the paradoxes which he considers: “they all result from a certain kind of vicious circle”(PM, i. 37). Russell states the error involved as followes: “vicious circle arise from supposing that a collection of  objects may contain members which can only be defined in terms of the collection as a whole…. More generally, given statements about any set of objects such that, if we suppose the set to have a total, it will have members which presuppose this total, then such a set cannot have a total”(PM, i. 37).  Afterward it is shown that to evade these paradoxes, Russell adopts a view that allows for ontological distinctions and then introduces a full-fledged theory of types in Principia Mathematica (Russrll, 1903: 500). Nevertheless, the new framework yields problems of its own that pose a threat to Russell’s object-centered metaphysics but also deprives him of handling truths of unrestricted generality. Then to present a final version of his Logicism, Russell’s way out of these issues will be set forth which comes in form of axioms of reducibility and infinity.

  1. 2.      Material

In this essay we’ve benefitted from Russell’s original and translated works that are concerned with the aforementioned period. Works by certain commentators of Russell’s philosophy have also been translated and drown upen accordingly.

2.1. Method

In the introduction, we have represented initial links between logic and mathematics to explain Russell’s logicism roots. Then we have classified Russell’s logicism into three periods. First period is about Russell’s idealistic views and his logicism that is based on this view. Second period is about Russell’s anti-idealistic views and his realistic metaphysics that is calld Platonic Atomism. And third period is about Russell’s moderated idealism and his modified logicism.

2.1.1.       Conclusion

Always Russell was influenced by his methaphysics for representing his own logicism. Obviously in Russell’s mind metaphysics have defined limits for logicism. For Russell as a logicion it was important to represent a version of logicism. He founded the foundations of the new logic. However his solution ultimately have been complicated but he opened the ways for new logic.

References

  1. Howard Whitley, Eves, (1953). An introduction to the History of Mathematics, 321, translated by Vahidiye Asl, University Publication Center.
  2.  Hylton, peter, (1990). Russell, idealism, and the emergence of analytic philosophy, 72, Oxford: Oxford University Press.
  3. Hylton, peter, (1990). Russell, idealism, and the emergence of analytic philosophy, 112-115, Oxford: Oxford University Press.
  4. Russell, Bertrand and Whitehead, A. N. (1927), Principia Mathematica to *56.  Cambridge: Cambridge University press.

کلیدواژه‌ها


عنوان مقاله [English]

Foundations of Russell’s “Logicist Thesis”

نویسندگان [English]

  • masoomeh AliHasanzadeh Asl 1
  • Masoud Omid 2
1 Graduate of philosophy, Tabriz University (corresponding author)
2 Associate Professor of philosophy department, Tabriz University
چکیده [English]

This paper is concerned with a general account of logicism as developed within Russell’s philosophy of mathematics. At first Russell’s approach in metaphysic was idealistic and he relied on this point of view for extending his platonic atomism. To expound this account it will be demonstrated that after developing platonic atomism, Russell attempted to present his logicism in The Principles of Mathematics as a view opposed to an idealistic account of mathematics. However, a number of paradoxes arose had their roots deep in Russell’s metaphysical views. At this point he was forced to moderate his realistic extreme approach. Afterward it is shown that to evade these paradoxes, Russell adopts a view that allows for ontological distinctions and then introduces a full-fledged theory of types in Principia Mathematica. Nevertheless, the new framework yields problems of its own that pose a threat to Russell’s object-centered metaphysics but also deprives him of handling truths of unrestricted generality. Then to present a final version of his logicism, Russell’s way out of these issues will be set forth which comes in form of axioms of reducibility and infinity. Although these axioms represent a solution to remove paradoxes, actually those are too complicated to use.

کلیدواژه‌ها [English]

  • logicism
  • philosophy of mathematics
  • set
  • foundation of mathematics
  • proposition
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