This material is taken from a note written to my brother Robin in July 2013.
I plan to incorporate the material to ‘Journey in Being’.
The universal metaphysics is secure and certain up to the point that proof of existence of the Void is taken as valid. This document examines the nature of objects whose existence is specified in terms of characteristics rather than by showing. This enables assessment of the higher regions of the universal metaphysics and brings the question of certainty there in line with certainty in significant endeavour in general, especially as in mathematics.
Much of what I present as ‘my system’ is not new (as an example the stuff about human / universal identity is seen in Indian Philosophy). ‘Limitlessness’ is known in philosophy as ‘the principle of plenitude’. However a variety of things is new. The general approach that I have not outlined here is one from the concept of ‘Being’ which makes the possibility of metaphysics as knowledge of things as they are (though not all things) actual and possible and this has been philosophically in doubt since the time at least of the Scottish philosopher David Hume and the German philosopher Immanuel Kant. This doubt was underlined by the school of Logical Positivism of the early twentieth century and which was influential on twentieth century attitudes (some of which you and I inherit as ‘common sense’ at least) which continue on even though Logical Positivism is no longer dominant. So that is a first thing that is new—demonstrating the possibility of metaphysics and developing an actual metaphysics. It is further interesting that I have taken metaphysical development starting with ‘experience’ which here is a name roughly equated to consciousness and Being (roughly a name for what there is without attaching labels such as matter, mind, process) up to the point just before the proof of the fundamental principle (‘limitlessness’) which is not insignificant and which, since it does not depend on the fundamental principle, is above all except perhaps the most stringent / neurotic doubt.
Another interesting feature is (coming down to practical knowledge) as in science which we do not know to be exact even in immensely precise cases such as the equivalence of gravitational and inertial and gravitational mass and quantum electrodynamics, I have been able to begin a reasonable account of such knowledge measured by other criteria and framed by the metaphysical developments above. Now if ‘identity’ and ‘plenitude’ are not new, what is new about them? First, the proof is new and as far as I can tell a far better proof than anything in previous metaphysics either modern or scholastic or Greek regarding these questions. Second the account (that I show and deal with the doubt which—remember—is not one of inconsistency within the system or of agreement with what we know empirically) is more open and perhaps honest (except of course that Plato was in many was an ideal of openness). On the way, my approach to science—detailed on the website—is also more open and reasonable than those philosophies and viewpoints that see science in light of scientific theory as universal—whether as proving the case or as accepting impossibility or essential tentativeness of the case.
I am not quite so sure how far this extends to mathematics but have made some progress in this direction as well. The first aspect of this progress is to see the principle of ‘limitlessness’ more clearly. It is equivalent to ‘Logic’ but this is not to be conflated with the usual notions of logic or the logics without clarification and amendment. However, from this form it follows that every concept (read system of propositions) that satisfies Logic has an object. This is a version of Platonism in mathematics but different from the common Platonism (Gödel subscribed to some version of Platonism) that sees mathematical objects as real but existing in some ideal ‘Platonic’ world; in the present version mathematical objects are real but exist only in the Universe (outside which there is nothing not even an ideal world—and this whole notion of idea versus material versus mental world etc. is one that needs to be worked out before this makes complete sense and the requisite working out has been done at least to some extent). Now if mathematical objects exist in the Universe (which as is customary in my writing I have begun here to capitalize) then where and what are they? If mathematical and other abstract objects are timeless etc. how can they reside in a temporal universe? Where they are may be answered following the suggestion that defines a number as a set of sets that may be placed in one to one correspondence with the appropriate set; a number is the set of collections of objects each of which has the same ‘number’ of objects (you may wish to refer to B. Russell’s discussion of this point for further clarification).
Now the modern philosopher and mathematician will or may object to this empirical definition of a mathematical object (just as there may be objection to the earlier definition of Logic as an empirical object) but the counter to that objection is to observe that mathematics—i.e. the systems of mathematics such as arithmetic and geometry—began empirically, went through an ideal / abstract phase (i.e. up to today), but is now here seen to return at a new level to the empirical which is an abstract rather than a get your hands dirty empirical and no less in stature than the ideal because it is equivalent to the ideal but further because its foundation is not shaky as is the Platonic Ideal (where do the Platonic ideals live and if we postulate them because of our strong mathematical intuition surely to be real they must reside in some place that is more than a ‘another’ world that must exist because of intuition but that we cannot locate) and finally because the metaphysical system I have developed shows that there is nowhere else that numbers and other mathematical objects can reside but in the one and only Universe. One question remains: what is the temporal status of these objects that we used to think of as ideal and as residing in some non-temporal ‘universe’? We (today’s thinkers) think of them as non temporal but what is here revealed is that they are not non-temporal; rather they begin with the ordinary empirical and temporal (here I am imagining that the Universe is temporal but in fact it is not entirely temporal but that is another question discussed in detail in my general essays but which I omit from this discussion) but temporality does not survive abstraction—that is when the abstract objects are non-temporal the non-temporality is contingent rather than necessary (the discussion might need modification to be worked out for the incomplete temporality of the Universe but I have not worked it out—I suspect that since the requirement for form at all is rudimentary identity, i.e. sameness, the working out would include terms of identity).
Now we are in a position to address the question of certainty in mathematics (and Logic) from the point of view of ‘my system’ (it has a name but I have not named it in this note). A more complete answer needs to be worked out (perhaps) but here is a first answer. Certainty is given in those structures that are open to complete inspection or those that have features that are open to complete inspection (or that are equivalent to such structures or features)—even if the entire system is or cannot be shown perhaps some part of it is or can be projected in such a way to be show-able (which might necessarily mean ‘finite’). In other cases certainty is not given; probably these other cases are significant and of interest. Though primitive, this is rather parallel to the case of science. If we regard a scientific theory as a universal hypothesis we can not know its truth until we know the entire universe (at least in essence as concerns the objects to which the hypothesis refers) (the situation is not symmetric with regard to falsity—we do not need to know the entire universe to know falsity of a hypothesis). On the other hand if we think of a theory as applying to a limited domain (e.g. the set of phenomena which it is contrived to explain) then it is complete and certain (over those phenomena) and this is not at all trivial from practical points of view even though not as pretty as universality.
Incidentally here is a new unification of science and logic; early in science Bacon sought an inductive inference for to infer scientific theories in science that would parallel deduction in logic; then it was recognized that science is generalization and logical process is not so the sought parallel is not reasonable; now, however, we see that what should be compared is—on one hand—deduction under science and deduction under logic and—on the other hand—inductive inference from data of scientific theories and induction from our descriptions of the world to logical systems such as the propositional and predicate calculi. Some of the foregoing are among my earlier thought but the thoughts regarding certainty in mathematics though not entirely new crystallized as I wrote above and of course are likely to be revised if / when I return to them.
You may know that Leopold Kronecker was critical of Georg Cantor’s work on set theory and infinity. Cantor, as you probably know, noted that a finite set cannot be placed in a one to one correspondence with a proper subset of itself. On the other hand the integers can be placed in 1-1 correspondence with the even integers. This was basis for understanding of Cantor’s notion of the infinite and particularly of ‘Aleph zero’ the ‘first infinite number’. Cantor went on to define operations for infinite numbers and other infinite numbers. Particularly, he showed the cardinality of the real numbers is 2** ‘Aleph zero’ (two raised to the power ‘Aleph zero’ where 2 raised to the power of any number is the number associated with the power set of the set associated with the number). The continuum hypothesis says that the cardinality of the reals is ‘Aleph one’ the next infinite number after ‘Aleph zero’. Kronecker’s criticism which you probably know was that we have not demonstrated the fact that the set of integers really exists but have merely shown it as a set subject to certain properties (selected with care etc. so as to make for arithmetic). Thus the infinities of Cantor should perhaps be thought of as ‘potential’ rather than actual. The situations in mathematics where certainty is not given (foregoing discussion) are those for which the ‘object’ is not shown but is understood via properties. This is significant because the situations for which mathematical objects can be actually shown is limited. However, because of the beauty and—especially—the power that comes from this admission Hilbert said ‘No one shall drive us from the paradise which Cantor has created for us’. (I know or believe that this is of course not the end of this story but I do not currently know enough to continue it.) There is some parallel between admitting the potential objects into mathematics and the admission into metaphysics of the part of metaphysics that comes after introduction of existence of the Void. There is always, as first shown in the paradoxes that shook the foundations of mathematics and that culminated with Russell’s paradox c. 1900, a potential for contradiction / paradox when some object is specified by properties rather than pointing out. In mathematics and logic care is required to eliminate potential paradox. Here, in the metaphysics of Being developed by me with inspiration from Wittgenstein and Heidegger and Indian Philosophy and others, paradox is blocked by implicit blocking at outset. In simple situations this is sufficient. I expect that in the complex workings of the system (in principle it admits of every existing object) paradox will have to be blocked by situation dependent devices.