Online research in philosophy

I argue that when empty space is seen in mirrors—that is, when perceptual specular experience is veridical—specular empty space is, like pictorial empty space, seen-in. I explain how the phenomenal expansiveness of specular reflections can nonetheless be reconciled with the see-through look of specular space.

This is a dialogue in the philosophy of mathematics. The dialogue descends from the confident assertion that there are infinitely many numbers to an unresolved bewilderment about how we can know there are any numbers at all. At every turn the dialogue brings us only to realize more fully how little is clear to us in our thinking about mathematics.

Humans possess two nonverbal systems capable of representing numbers, both limited in their representational power: the first one represents numbers in an approximate fashion, and the second one conveys information about small numbers only. Conception of exact large numbers has therefore been thought to arise from the manipulation of exact numerical symbols. Here, we focus on two fundamental properties of the exact numbers as prerequisites to the concept of EXACT NUMBERS : the fact that all numbers can be generated by a successor function and the fact that equality between numbers can be defined in an exact fashion. We discuss some recent findings assessing how speakers of Munduruc (an Amazonian language), and young Western children (3-4 years old) understand these fundamental properties of numbers.

Abstract A number is the number of a class which is an objective, nonactual, mathematical object. The concept of class is analyzed and it is concluded that a number is the number of a pure founded class. A tempting strategy of explaining numbers away is rejected. Some well?known definitions of numbers are analyzed and it is concluded that this analysis purports the thesis that the unique notion of number does not exist. Numbers are conventional. Nevertheless, an argument is offered purporting the thesis that von Neumann's ordinal numbers are the ordinal numbers. Accordingly, the corresponding von Neumann's cardinal numbers are the numbers.

A number is the number of a class which is an objective, nonactual, mathematical object. The concept of class is analyzed and it is concluded that a number is the number of a pure founded class. A tempting strategy of explaining numbers away is rejected. Some well-known definitions of numbers are analyzed and it is concluded that this analysis purports the thesis that the unique notion of number does not exist. Numbers are conventional. Nevertheless, an argument is offered purporting the thesis that von Neumann's ordinal numbers are the ordinal numbers. Accordingly, the corresponding von Neumann's cardinal numbers are the numbers.

In this paper we define n+1-valued matrix logic Kn+1 whose class of tautologies is non-empty iff n is a prime number. This result amounts to a new definition of a prime number. We prove that if n is prime, then the functional properties of Kn+1 are the same as those of ukasiewicz's n +1-valued matrix logic n+1. In an indirect way, the proof we provide reflects the complexity of the distribution of prime numbers in the natural series. Further, we introduce a generalization K n+1 * of Kn+1 such that the set of tautologies of Kn+1 is not empty iff n is of the form p , where p is prime and is natural. Also in this case we prove the equivalence of functional properties of the introduced logic and those of n+1. In the concluding part, we discuss briefly a partition of the natural series into equivalence classes such that each class contains exactly one prime number. We conjecture that for each prime number the corresponding equivalence class is finite.

For first order languages with no individual constants, empty structures and truth values (for sentences) in them are defined. The first order theories of the empty structures and of all structures (the empty ones included) are axiomatized with modus ponens as the only rule of inference. Compactness is proved and decidability is discussed. Furthermore, some well known theorems of model theory are reconsidered under this new situation. Finally, a word is said on other approaches to the whole problem.

Numbers -- Hallucinations -- Fictions -- Scientific languages, ontology, and truth -- Truth conditions and semantics.

If everything exists, then it looks, prima facie, as if talking about nothing is equivalent to not talking about anything. However, we appear as talking or thinking about particular nothings, that is, about particular items that are not among the existents. How to explain this phenomenon? One way is to deny that everything exists, and consequently to be ontologically committed to nonexistent “objects”. Another way is to deny that the process of thinking about such nonexistents is a genuine singular thought. The first strategy we may call “the Meinongian tradition” (championed by authors like Alexius Meinong, Ernst Mally, Terence Parsons, Richard Routley, and Ed Zalta), while the second could be dubbed “the de re tradition” (connected to work by Gareth Evans, John McDowell, and Tyler Burge). Finally, the third way to solve the above puzzle, and probably the majority view in contemporary philosophy, is due to Bertrand Russell and W.V.O. Quine, who deny the particularity of the apparent nonexistent object and the singularity of the corresponding thought via the view that any statement about apparently particular nonexistents can be paraphrased into a quantified expression containing no genuinely referring terms. Jody Azzouni’s book is an attempt to argue for and develop a fourth view, based on the hitherto unrecognised notion of an “empty singular thought”, which Azzouni takes to have a place in logical space. Concomitant to developing the view, Azzouni applies it to three typical cases of talk about nonexistents: numbers, hallucinations, and fictions. As the name suggests, empty singular thought is devised as having three essential characteristics: (1) it is genuine thought, no different from any other, (2) it is singular, that is, its content is partly determined by particular non-conceptualised states of affairs, and (3) nevertheless it is genuinely empty, unlike Meinongian thought, that is, its object “does not exist in any sense”, to use Azzouni’s own formulation. Azzouni undertakes some challenging acrobatics when trying to persuade the reader that his view is substantive and it does not end up being the same as any of the previous three views about apparent talk about nonexistents..