By Tuvshe Bold

      The second necessary element (first being quantitative property) for interpretation of mathematical concepts is man's ability of to abtstract. By abstraction, I do not mean pure imaginary creation like the one intuitionists propose. Instead, I mean the mind's ability to abstract the quantitative property from bodies and use it without the presence of bodies. Explanation of exactly how the abstraction works belongs to psychology. For the purpose of this paper, it is sufficient to notice that I can add without witnessing addition. The process of addition is possible by, first, recognition of quantitative property and secondly, abstraction of it.

      The science of mathematics finds itself necessarily and essentially based on abstraction, since mathematicians never empirically test, for example, putting 200 mice and 200 mice to create 400 mice. The fact that all of mathematics is abstract, I believe, was one of the motives of intuitionists to think mathematics is a sole product of the mind. However, their inference that because we do not empirically prove a mathematical statement, it is only a figment of imagination, was wrong.

      In addition to quantitative property and abstraction, a third important element is as equally important as the concept of infinity; infinity is based on the concept of possibility. I will call it "possibleness" to distinguish from use of the word "possibility." There are infinite natural numbers, because, however large a number is, it is possible8 for there to be a larger one.9 Possibleness applies similarly to divisibility as it does to addition. For every single number, at any instant, it is possible to divide that number. Consequently, just as there is no biggest number, there is no last possible stage where further divisibility is impossible. One should be careful to not take possibleness as granting that there are infinite numbers of entities in the world. All possibleness grants is the possibility of another step of addition, division, multiplication and etc. Any finite number, as was determined previously, is an alternative name for another quantity. It is a quantity itself too. 1099 is simply another way to say 1, 1,...(1099-31s)..., and 1. Whether there exists 1099 bodies in the world or not is an irrelevant issue here because we do not need 1099 bodies to use 1099. Initially, all we need is one body and, through abstraction, we can get 1099the same way we get 2 or 3. Once we posses the concept of one body, we do not need any body at all to understand 1099. That is why I used "initially."

      Infinity is not a quantity, but a concept based on unrestricted possibility. Infinity is a character of possibility. It is common to confuse infinity as quantity and produce language such as "infinitely large number", which is a mixture of possibleness and quantity. There is no such thing as an "infinitely large number." Though, the phrase "infinitely large set" can be described as characterization of a set consisting of every single possibility which is endless. Therefore, what is meant by "infinitely large" is simply "endlessly possible" and what is meant by "infinitely large number" is the possibility of having a larger number at any instant. Again, for example, consider the set of all numbers divisible by 2. It is an infinitely large set because the possibility of finding even numbers is endless. On the other hand, even though we conceive infinity and a large number through the same way, no matter how large a number is, it is a quantity. The present explanation of infinity gives a nice way out for Russell's axiom of infinity, one of the two major problems of the logicists' program: "There are infinite number of things in the universe." This axiom was rejected because it is not logically sustainable. All we need ito notice is that it is possible for there to be infinite number of things in the universe. Furthermore, as far as formalists and intuitionists are concerned, they reject infinity because we cannot "survey" it. However, "infinitely large number", by definition, cannot be surveyed, for it is not a quantity but a concept of possibility. The intuitionists made the common mistake of confusing concepts and their meanings.

      Lastly, after integers and infinity, the concept of fraction is left to be explained. As it turns about, it is very simple. The concept of fraction is just based on what I defined before as abstraction and possibleness. Consider the fraction number 0.33. The concept of this number consists of concepts of 0 and 3, and the possibility having a .33rd of the whole number 1. The fraction .334 denotes another possibility and so does .333 and etc. The number (3.14159....) denotes on possibility and 3.24159.... denotes another, whereas 3.1415(9) denotes a different possibility. The issue involved with rational and irrational numbers is completely irrelevant for interpretation of concepts of fraction as Arend Heyting is overly concerned with the difference in his [The intuitionist foundation of mathematics]. The mistake comes from the fact that it is common sense to think of any non-integer number as expressible in the n/p form. As far as mathematical concepts are concerned, rational numbers as n/p and irrational numbers as q are just a matter of different ways of expression. The difference between them is issue within mathematics to be explained by mathematical terms and language.

      I have now given an account of mathematical concepts of integers, infinity and fractions, and of mathematical certainty. What have been used to describe these concepts can be easily used to describe any other mathematical concept including negative numbers, complex numbers, calculus and etc. To sum up, quantitative property, abstraction and possibleness are said to be the three building blocks of philosophy behind mathematical concepts. I examined the defects of the intuitionism, formalism and logiscism and offered an alternative view that solves all the defects. Here, the logicists deserve a little extra space for consideration. Their task concentrated on mathematical propositions, not on mathematical concepts. They did have theories for mathematical concepts, but those theories were constructed as necessary to work for their main goal: deduction of mathematical theorems from logical axioms. And thus, I should not like to say their account had defects, for they were not trying to give explanation of mathematical concepts, but had a different purpose.

      Given the solution in this paper, there still remains a great deal of study on the subject that needs to be investigated with greater detail. Study of the nature of mathematical propositions probably promises the most excitement. I strongly believe the logisticians' method is the sole correct way to pursue this issue. If not exactly the same, essentially it would probably be a similar formal system. Moreover, there are issues, such as how our perception of abstraction or of possibility works that need detailed psychological analysis. Explanation of geometry and the relationship between geometry and mathematics is another issue that is very interesting.