المنهج العلمي الواقعي | علي آل شُبَّر

The fact that ‘being an ω-sequence’ is purely structural, and that the numbers taken as a whole instantiate this structure, does not imply that individual numbers are purely structural. Numbers are places or ‘offices’ in structures but not merely places in structures. The real relations between heaps and unit-making properties – the true numbers – taken as a whole system, form an ω-sequence, but still retain their own characters. Other entities, such as sets formed out of the empty set, can also form an ω-sequence, without anyone needing to mistake them for numbers. That would be like mistaking the commuters in the queue for the 8.21 bus to the City for those in the queue for the 8.46, simply because both are queues. An Aristotelian Realist Philosophy of Mathematics | James Franklin

For the continuum seems to contain all there is to ratios: the system of all ratios, with their mutual relations of closeness to one another, just is the continuum. Constructing it in set theory does show its purely structural nature, but at the cost of a great deal of unintuitive machinery. Granted however that the continuum is purely structural, it might seem to follow that ratios (and hence quantity in general) are purely structural. Identifying the flaw in this argument casts light on a number of problems of the relationship between structural and non-structural properties, and hence on many issues concerning the applicability of mathematics to reality. It is true that the continuum is a purely structural entity. It does not follow that the individual ratios themselves are purely structural. The continuum is a structure which the system of ratios shares with several other quite different entities, for example, an infinite line in real space (if real space is infinite and infinitely divisible), and the set of infinite decimals (with distance defined by the difference). It does not follow that either individual spatial points or individual infinite decimals are purely structural. Only the system of all the entities of each kind instantiates the (same) structure, the continuum. An Aristotelian Realist Philosophy of Mathematics | James Franklin

It was explained in the previous chapter that the ontological commitments of set theory and of mereology (with individuals) were the same, as sets supervene on objects and the existence of unit-making properties for them, which are the same entities needed for a ‘calculus of individuals’. ... It follows that to demonstrate that a concept is purely structural, it is sufficient to construct a model of it out of sets – the capacities of set theory and pure mereology for construction are identical. That is indeed the main philosophical point of the construction of mathematical entities in set theory. While construction in set theory is normally taken to support Platonism, by reducing the objects of mathematics to sets considered as Platonic entities, it can equally be read as mereology and structure in disguise. An Aristotelian Realist Philosophy of Mathematics | James Franklin

Since equality and inequality of objects are engendered by the repeated instantiation of a unit-making universal, this confirms the lesson of the previous chapter, the crucial role of unit-making universals in generating numbers and sets. Equality is admitted into the logic for the same reason as that the mereology being used is the ‘calculus of individuals’: there must be objects constituted as units. An Aristotelian Realist Philosophy of Mathematics | James Franklin

Modern pure mathematics has concentrated more and more on pure structure. Poincaré recognized the new direction of mathematics in his celebrated comment:

Mathematicians do not study objects, but the relations between objects; to them it is a matter of indifference if these objects are replaced by others, provided that the relations do not change. Matter does not engage their attention, they are interested in form alone.

That comment is not especially true of mathematics conceived as the science of quantity, but it is true of the higher mathematics of 1900 and since. An Aristotelian Realist Philosophy of Mathematics | James Franklin

[تفسير الحال والواقع] قد يظن بعض القاصرين عن النضج العلمي أن المدارس المخالفة لما يميلون إليه لا صدق فيها، وإنما هي عبارة عن مدارس الكذب المحض. وفي الحقيقة هذا يكاد أن لا يكون في المجتمع العلمي، حتى في العلوم الأسوء حالًا بلحاظ الفلسفة، مثل الرياضيات، فإن جميع المدارس الرياضية الفلسفية المشهورة والمتداولة فيها حق مخلوط بالباطل. وقد تتميَّز بعض المدارس بحق تتفرد به، ولكن هذا الحق المتفرد به ممزوج بأمر باطل متفرد به أيضًا. وهذا قد يستلزم مشكلة من جهتين: الجهة الأولى: أن الذي يُدرك تميُّز هذه المدرسة بذلك الحق، فيميل إليها بسبب الحق الموجود فيها خصوصًا دون سائر المدارس، قد يأخذ حقّه هذا ممزوجًا بالباطل معتقدًا المخلوط. وأزيَد من ذلك فإنه قد يأخذ ما بالعرض مكان ما بالذات، فيظن أن الذي هو سبب التميز العلمي لمدرسه ليس مجرد الحق بل بمدخلية ذلك الباطل وامتزاجه به. الجهة الثانية: أن الذي يدرك تميُّز هذه المدرسة بذلك الباطل، قد ينفر عنها نفورًا كليًّا، فلا يطالعها أصلًا، فيغيب عنه ذلك الحق الذي تفردت به هذه المدرسة. فإذا كان الأمر بهذه الصورة ليت شعري ماذا نفعل حتى نتلبس بالحق ونتجرد عن الباطل؟! الجواب: إن الإنسان بمفرده قد لا يحصّل الحق المطلق في جميع المبادئ والمسائل، ولكنه بمعاونة الآخرين قد يقترب من ذلك أو يحصّله. بعبارة أخرى: ذلك بالاستعانة بالحليف والخصم معًا، وإن تعصّب الحليف، وإن أبى الخصم، حتى يرتقي فوق الحليف والخصم. بعبارة صريحة: بقراءة كتب جميع المدارس المعتبرة عند علماء التخصص، فيعرف الباطل في كل مدرسة ويعرف الحق في كل مدرسة، فيترك الباطل ويجمع الحق.

[ترجمة مختصرة لمؤلف الكتاب] جيمس فرانكلن عالم رياضيات أسترالي حصل على الدكتوراه في الرياضيات من جامعة وارويك [Warwick] (عام 1981 م) في الـ(Algebraic Groups). ثـم تعين أستاذًا في قسم الإحصاء والرياضيات في جامعة (New South Wales) من عام 1982 حتى تقاعد عام 2019. أسس مدرسة سيدني في فلسفة الرياضيات، وهو فيلسوف واقعي (إن شئت أرسطي أو مشائي)، قد جمع التخصص في الرياضيات (دكتوراه في الرياضيات المحضة) مع الفلسفة. ألّف كتبًا كثيرة منها: — The Science of Conjecture: Evidence and Probability Before Pascal. — Corrupting the Youth: A History of Philosophy in Australia. — What Science Knows — An Aristotelian Realist Philosophy of Mathematics. — The Worth of Persons: The Foundation of Ethics. — The Necessities Underlying Reality.

[توصية بكتاب في فلسفة الرياضيات] عنوان الكتاب: 'فلسفة واقعية أرسطية للرياضيات' المؤلف: جيْمس فرانكلن قلت: هذا كتاب في فلسفة الرياضيات مبني على المنهج العلمي الواقعي، وبالتحديد [وبحسب تعبير المؤلف] مبني على الفلسفة الأرسطية الشبيهة بالأفلاطونية. والكتاب أحسبه يفيد جملة كبيرة من القراء، ومنهم: — من بدأ مسيرته الرياضية [البرهانية لا الأداتية الحسابية العملية] بالرياضيات القديمة ولم يعرف فلسفتها الواقعية. — من بدأ مسيرته الرياضية بالرياضيات القديمة وعرف فلسفتها الواقعية، ولكنه لم يبدأ بعد بالرياضيات الحديثة، أو بدأها ولم يعرف فلسفتها. — من هو رياضي بالتخصص الجامعي بغض النظر عن درجته العلمية (بكلريوس، ماجستير، دكتوراه). وذلك لأن الرياضي غالبًا لا يدرس فلسفة الرياضيات بنحو علمي جاد، وإن اتفق أن طالعها أو درسها فإنه غالبًا يدرس الفلسفة المثالية الأفلاطونية، أو الفلسفة الإسمية [التي منها المذاهب الصورانية المختلفة والمذاهب المنطقية]. — وبالجملة: كل من هو في منزلة بيرتراند راسل في التعقل، وهؤلاء يمثلون مجموعة كبيرة من المجتمع الرياضي. إذ قال بيرتراند راسل: "الرياضيات قد تعرَّف بأنها الموضوع الذي لا نعرف أبدًا عن ماذا نتكلم عنه، ولا عن صحة كلامنا" فكل من وجد كلام بيرتراند راسل منطبقًا عليه فإنه يستفيد من هذا الكتاب بكل تأكيد. ____ ملاحظة: إن انتهيت من انتخاب ونشر النصوص التي تستحق النشر من هذا الكتاب، سأكتب مراجعة عامة مع نشر بعض التعليقات في القناة. ثـم وقد يكون من المتعارف، ولكنني أنبه على أن نشر النصوص لا يستلزم بالضرورة الاعتقاد بمضمونها.

The relation between discrete and continuous quantity may be clarified by asking: is ‘being one kilogram mass’ a unit-making property in the sense of the previous sections on discrete quantity? The answer is: yes and no. On the one hand, a unit such as one kilogram is subject to repetition, so in that sense resembles a discrete entity such as an apple. On the other hand, ‘being one kilogram’ contrasts with ‘being an apple’ in being subject to continuous variation: being 1 kilogram is an arbitrary point in a range of indefinitely close weights, but there is no quantity of apples between one and two apples. (Although I may eat one and a half apples, the half is a quantity of apple-stuff which is half that of a typical apple either by weight or volume.) An Aristotelian Realist Philosophy of Mathematics | James Franklin

So, a set refers to the division of a heap into objects in abstraction from which unit-making properties are doing the dividing. Thus the view that numbers are properties of sets is somewhat more abstract than the claim that they are relations between a heap and a unit-making property, but compatible with it. An Aristotelian Realist Philosophy of Mathematics | James Franklin

Philosophers sometimes speak as if sets were only discovered in the nineteenth century in connection with esoteric questions in the foundations of real analysis.An explicit set theory was only needed at that stage, but there is plenty of reference to sets in earlier mathematics. An Aristotelian Realist Philosophy of Mathematics | James Franklin

Whereas ratios have nothing to do with sets, numbers are intimately connected with them. Given a set, there is something to count. And conversely, if there is counting, there is a set of entities being counted, and indeed sets are good for little else. Given a heap and a unit making property structuring it, there is immediately created (there supervenes) both the set of things of which the heap is the mereological sum, and the number of things in that set. If there is no unit-making property – if there is just stuff – there is no number and no set. An Aristotelian Realist Philosophy of Mathematics | James Franklin

Like a ratio, a number is not merely a position in the system of numbers. There is a perfectly determinate number of apples in a heap, independently of anything systematic about numbers (and independent of any knowledge about it, such as through counting). Thus the theoryof Shapiro and others that the number 2 is merely a position in the series of numbers is incomplete. It arises from a confusion between the ordinal and cardinal aspect of numbers. Although 2 as an ordinal number is merely the second place in the number sequence, 2 as cardinal number is the unique relation between, for example, Sirius and ‘being a star’ (since Sirius is a double star). An Aristotelian Realist Philosophy of Mathematics | James Franklin

...numbers are not properties of parts of the world simply, but must be properties of the relation between parts of the world and the unit-making properties that structure them. So the fact that a heap of shoes stands in one such numerical relation to ‘being a shoe’ and another numerical relation to ‘being a pair of shoes’ (made much of by Frege) does not show that the number of a heap is subjective or not about something in the world, but only that number is relative to the count universal being considered; and Aristotelians take that universal to be part of the world’s furniture. An Aristotelian Realist Philosophy of Mathematics | James Franklin

Ratios appear to have no close connection with sets. The ratio of your height to mine does not suggest or require the existence of any sets (of heights, people, numbers or anything else). An Aristotelian Realist Philosophy of Mathematics | James Franklin

A (particular) ratio is thus not merely a ‘place in a structure’ (of all ratios), for the same reason as a colour is not merely a position in the space of all possible colours – the individual ratio or colour has intrinsic properties that can be grasped without reference to other ratios or colours. Though there is indeed a system or space of all ratios or all colours, having its own structure, it makes sense to say that a certain one is instantiated and a neighbouring one not. It is perfectly determinate which ratios are instantiated by the pairs of energy levels of the hydrogen atom, just as it is perfectly determinate which, if any, shades of blue are missing. An Aristotelian Realist Philosophy of Mathematics | James Franklin

The position that will be argued for here is that quantity and structure are different sorts of universals, both real. The sciences of them are approximately those called by the (philosophically somewhat unsatisfying) names of elementary mathematics and advanced mathematics. That is a more exciting conclusion than might appear. It means that the quantity theory will have to be incorporated into any acceptable philosophy of mathematics, something very far from being done by any of the current leading contenders. It also means that modern (post-eighteenthcentury) mathematics has discovered a completely new subject matter, pure structure, thus creating a science unimagined by the ancients. An Aristotelian Realist Philosophy of Mathematics | James Franklin