Notes on Plato: mathematical entities, i.e. Forms = (point, line, circle,..) as a direct description
> not a direct description of the perceived, certain valuable ideas, especially these three, i.e. point, line, circle,
“One of the penalties for refusing to participate in politics is that you end up being governed by your inferiors.”
“Let parents bequeath to their children not riches, but the spirit of reverence.”
Notes on Aristotle: forms are just part of objects, Forms do not have an autonomous existence detached from
the objects, no “actual” infinity (but rather “potential”),
> not a direct description of the perceived,if forms are not detachable, then there might exist certain configurations that are possible and those are not, step forward with “parts”, indicates “incompleteness” of the perceived, but “no autonomous existence” of mathematical objects should concern the perceived only,
“The aim of art is to represent not the outward appearance of things, but their inward significance.”
“Suffering becomes beautiful when anyone bears great calamities with cheerfulness, not through insensibility but through greatness of mind.”
Notes on Leibniz: truth: of fact and of reason (contingent), propositions do not have “mathematical” content (like before), but are “necessarily true”,
> assumptions that the models used beneath the theorms are not contradictory is impossible, step backwards as any non-perfect description of the perceived might be contradictory with other theorems in the model, as well as the perceived itself might be contradictory with other entities within the perceived,
I do not conceive of any reality at all as without genuine unity.
There are also two kinds of truths: truth of reasoning and truths of fact. Truths of reasoning are necessary and their opposite is impossible; those of fact are contingent and their opposite is possible.
Notes on Kant: analytic and syntethic (empirical or a posteriori) propositions in mathematics, synthetic a priori true if anything true (1 exists, 1 = 1, etc.), not dependant on the perceivable (in the context of Goedel), distiction between mathematical concepts and objects (only allowed due to the specific structure and behavior of the time and space), pure geometry and arithmetic are synthetic a priori, confirmation of Aristotle’s division between the potential and actual infinities, infinity neither perceptible nor constructible but not an impossibility, geometry allows at most logical analysis of the perceived space, mathematical theorems are a priori truths
> pure geometry and arithmetic may be synthetic a priori descriptions of the perceived, in fact, might be misleading, confirmation of the claim of no “mathematical content” within the mathematical proposition has ceased to blossom, with the distinction of the analytic and synthetic; still, synthetic elements may only refer to the perceived, ie. be the perceived synthetic a priori, i.e. pure mathematics describing the structure of space and time always decides about the perceived time and space, i.e. cannot be free from an empirical material;
All the interests of my reason, speculative as well as practical, combine in the three following questions: 1. What can I know? 2. What ought I to do? 3. What may I hope?
It is beyond a doubt that all our knowledge that begins with experience.
Notes on Dedekind: a notion of number independent of space and time, “coressponding to a thing”, “representing on thing by a thing”, “relate things”, arithmetic may be derived from logic,
> we might represent a perceived notion, based on the coresspondences and representations, we might talk about numbers, having in mind that these numbers at most only exist in the perceived,
Notes on Russell: mathematics and logic are identical, vicious circle principle, ramified theory of types, the definition of order, own axiom of infinity,
> can we logically derive numbers from logic? what other ideas besides the VCP might be used for throwing out the weaknesses in the models? on the contrary, does there exist a way to deduce about the perceived? an approach to eliminate the contradictions rather than more generic approach? axiom of infinity that connects quantitative values with pure logic – not a good idea?
Notes on Ramsey: elimination of quantifiers in definitions,
> as for the elimination of the quantifiers, there are some sentences like “all primes are prime” which correctly make use of “all”, but in general cases we should be more specific about the object; deriving mathematics from artificial axioms of infinity and reducibility, which represent at most contingent truths,
Notes on Hilbert: axiomatic approach to Euclidean geometry, introduction of geometrical model in real numbers, categoricity (having exactly one model up to isomorphism), the same opinion on geometry as the one of Kant, rigorized axiomatic method, mathematics can be founded on logic as much as the other sciences, grounding mathematics on spatiotemporal configurations, concrete signs, there is a place for infinitistic mathematics, but in mathematics restricted to finite concrete objects, the aim not to reduce mathematics to logic but representing its form using the concrete symobolism, constituents of mathematics are finitistic, formal representation system devoid itself of meaning, finitiary way of dealing with the finitistic,
> how to address the perceived spatiotemporal measures? infinistic in a sense of Kantor’s work only? what was his view of the infinity? belief in constistency of logic at given level of rigour later weakened by Goedel’s work, foundations for the work of Goedel on the finitistic, but not entire concrete, mathematics,
The further a mathematical theory is developed, the more harmoniously and uniformly does its construction proceed, and unsuspected relations are disclosed between hitherto separated branches of the science.
Notes on Brouwer: mathematical theorems are a priori truths, arithmetic should derive from the intuition of time, a constructive proof being a thought experiment, ““2 + 2 = 3 + 1” must be read as an abbreviation for the statement “I have effected the mental construction indicated by ‘2 + 2’ and ‘3 + 1’ and I have found that they lead to the same result.” “, “personal” mathematics, subjective, transsubjective, constructive admission, something being constructed, rejection of the excluded middle,
> The idea of having something constructed is somewhat foggy, might be true that certain objects cannot be created, reverse-engineering of logical operators are crucial, some objects may not exist, would learning about those require grasping more about the given?
Notes on Goedel: real propositions provable by ideal means which cannot be proved by concrete means, i.e. consistency of arithmetic cannot be proved by concrete means, attempts at establishing perfect consistency with the mathematics impossible, idea to streghen the finitistic mathematics through increasing the domain of objects to enable demonstration of consistency of arithmetic, concept of provability for the constructive logical operators, method of arithemetization, formal language, formal system, provability, decidability, non-logical axioms, truth complete, primitive recursive, product-of-prime representation, proof numbering, work within the system to obtain information about the system, “it is conceivable that there exist finitary proofsthat cannot be expressed in the formalism”
> some deduct that mind is not mechanichal due to incompleteness theorem (Penrose), Hilbert attempts at formalization of reasoning in formal systems, no unsolved problem shown to be independent of the Peano Arithmetic, what axioms we should accept and why?
Notes on von Neumann: peano already uses all that can be done in a finitiary way (Goedel changed his opinion after years, Hilbert has never agreed with that),
There’s no sense in being precise when you don’t even know what you’re talking about.
Anyone who attempts to generate random numbers by deterministic means is, of course, living in a state of sin.