طبق تلقی براون از افلاطونگرایی، که در اینجا به آن «افلاطونگرایی جدید» میگوییم، ماهیت ریاضیات در قالب هفت مدعا قابل صورتبندی است: واقعگرایی، تجرّد، جزئیت، شهودمندی، پیشینیبودن، خطاپذیری، و توسعهپذیری. در این مقاله تلاش شده است تا این دیدگاه، بر اساس دو معیاری که خود براون لحاظ کرده است، یعنی مقبولیت اجتماعی و مقبولیت روششناختی، نقد و ارزیابی شود. میزان مقبولیت اجتماعی یک نظریه را میتوان از روی فراوانی اشخاص موافق با آن سنجید. اما مقبولیت روششناختی یک نظریه به توانایی آن در حل بهینۀ مسائل مربوطه بستگی دارد. بهزعم براون، افلاطونگرایی جدید از هر دو لحاظ بهترین نظریۀ فلسفی دربارۀ ماهیت ریاضیات است. در این مقاله ضمن تقریر دقیقتر این دیدگاه، آشکار میشود که به لحاظ مقبولیت نهادی میتوان با براون همرأی بود. اما مقبولیت روششناختی آن همچنان مناقشهآمیز است، زیرا دستکم دو مسئلۀ دسترسی و مسئلۀ یقین که گریبانگیر افلاطونگرایی اولیه بود در افلاطونگرایی جدید نیز همچنان حل نشده باقی مانده اند. تأکید این مقاله بخصوص بر مشکلات دیدگاه براون در مواجهه با مسئلۀ دوم است.
تازه های تحقیق
A Criticism and Evaluation of the Modern Mathematical Platonism
PhD in Philosophy of Science, Islamic Azad University, Tehran. Iran,
This paper criticizes and evaluates the New Platonism, according to two major criteria: the social acceptability, and the methodological acceptability. The social acceptability of a theory, according to my definition, is the interest and attitude of the people to that theory; and it can be measured on the frequency or percentage of interested parties. But the methodological acceptance of a theory means to match it with criteria such as consistency, simplicity and explanatory power; and its value can be assessed based on its success in solving philosophical problems related to it. According to Brown, the new Platonism, is the best philosophical theory about the nature of mathematics, both sociological and methodological. As for sociological criteria, we can be sympathetic and agree with Brown. That is, it seems that the new version of Platonism is still acceptable. But it needs to prove its methodological acceptability. Because the access problem and the certainty problem are still not resolved.
Key Words: New Platonism, the access problem, the problem of certainty, the social acceptability, the methodological acceptability.
- 1. Introductıon
In the second part of the paper, I am rewriting Brown's account of the new Platonism, in which his main claims are sufficiently explicit and clear and distinct. The emphasis on these three features is because Brown himself has not paid enough attention in shaping his perspective. For example, he does not distinguish between types of realism, or types of errors, or the meaning of the particularity is clearly not expressed. After the introduction of New Platonism, in the third part, I will criticize this version of Platonism.
- 2. Brown's account of New Platonism
According to Brown's account, the nature of mathematics is in the form of seven claims to be formulated:
- Realism: The mathematical objects, properties and relations are perfectly real, that is, they exist first, and secondly, their existence is independent of us.
- Abstraction thesis: Mathematical objects are abstract, that is, they are not spaciotempral and secondly, they have no causal power.
- Particularity thesis: Mathematical objects, while being abstract, are particular.
- Intuition thesis: Mathematical entities can be ‘seen’ or ‘grasped’ with ‘the mind’s eye’.
- A priority thesis: Math is a priori, that is, it is not based on sensory data.
- Fallibilism:The argument presented for a mathematical claim may be an error.
- Extensibility thesis: the set of mathematical research patterns and techniques is not limited to formal proofing.
3. Assessing New Platonism
The two criteria that Brown himself has implicitly used in assessing New Platonism are the inclusiveness of the mathematicial community and its efficiency in solving the relevant philosophical problems. I call these two, respectively, institutional and methodological acceptability.
3.1. The institutional acceptability of The New Platonism
Brown and many philosophers argue that no theory is as popular as mathematicians as Platonism. It seems that the natural tendency of mathematicians is pure and applied to Plato. Brown refers to quotes from Hardy, Littlewoods, and Godel. The evidence is very much in favor of their claim. Many other experts, such as Hermann Weyl, Davies and Hersh, and many more. In addition, according to Dan Monk's statistical survey, the mathematical world is populated with 65% Platonists, 30% formalists, and 5% constractivists.
3.2. The Methodological Acceptance of New Platonism
Based on what we said about the meaning of methodological acceptance and its method of measurement, we must determine problems that matter to mathematics philosophers and then we can measure the new Platonism in their solution. Brown himself claims that Original (initial) Platonism is incapable of solving two important problems, and instead of the new Platonism they are able to solve both of them.
3.2.1. The problem of access: The Original Platonism with the classical theory of knowledge, which implies the causal relationship between the degree and the degree, is in conflict. Because it involves acceptance of abstract notions, and this type of entity has no causal power.
The problem of certainty: Original Platonism is contradictory with the fallibility of mathematical claims. Because Platoism requires a priority of mathematical knowledge, and a priority means “independence of experience”. therefore a newer or more experience can not lead to the rejection of any mathematical reason or claim.
Brown has attempted to solve these two problems, respectively, by rejecting the claim that the knowledge is causal and proving the possibility of intuitive justification in mathematics. For this, he used, the counterexample of the EPR experimentation and the analogy of the "the mind's eye." But his attempts are not successful. Because, firstly, the EPR does not violate the theory of causal knowledge, but only requires that scientific knowledge is not necessarily explained by the theory of causal knowledge. Secondly, Brown has not provided good reason for the existence of "the mind's eye" as a faculty.
- Brown, J. R. (2008). Philosophy of Mathematics: A Contemporary Introduction to the World of Proofs and Pictures, Routledge.
- Brown, J. R. (2012). Platonism, Naturalism, and Mathematical Knowledge, New York and London: Routledge.
- Monk, J. D., 1970, “On the Foundations of Set Theory”, American mathematical Monthly. 77, pp. 703-71.
- Stiner, Mark, (1973). “Platonism and the Causal Theory of Knowledge”, Journal of Philosophy, 70/3: 57-66.
عنوان مقاله [English]
A Critical Review of the Modern Mathematical Platonism
Some mathematical philosophers believe that we can achieve a new and better version of mathematical Platonism, by eliminating defects of original Platonism. According to Brown's version of Platonism, that here we call it “Modern Platonism”, the nature of mathematics can be formulated in these seven theses: realism, abstraction, particularity, Intuitiveness, priority, fallibility, and extensibility.
This paper criticizes and evaluates the New Platonism, according to two major criteria: the social acceptability, and the methodological acceptability. The social acceptability of a theory, according to my definition, is the interest and attitude of the people to that theory; and it can be measured on the frequency or percentage of interested parties. But the methodological acceptance of a theory means to match it with criteria such as consistency, simplicity and explanatory power; and its value can be assessed based on its success in solving philosophical problems related to it.
According to Brown, the new Platonism, is the best philosophical theory about the nature of mathematics, both sociological and methodological. As for sociological criteria, we can be sympathetic and agree with Brown. That is, it seems that the new version of Platonism is still acceptable. But it needs to prove its methodological acceptability. Because the access problem and the certainty problem are still not resolved.