We define the empty set as the set that contains no elements, but is there more than one empty set? So is there "an" empty set as opposed to "the" empty set? May one be able to receive values, while another is truly empty, etc.? And how is it possible to define the empty set by the absence of members or by emptiness?
For a long time I have been very concerned with clarifying mathematics, primarily for myself but also because I plan to teach. After decades of reading and questioning and thinking, it seems to me that the philosophical views of mathematics are nonsensical. What does it MEAN to question whether mathematical objects exist outside of our minds? It sounds absurd. It seems clear to me that mathematics is a science like all the others except that verification (confirmation) is different. It is the science of QUANTITY and its amazing developments and offshoots (like set theory). And all sciences are products of our minds. They are our constructions, as are most of the physical objects in our immediate worlds. Shoes, sinks, forks, radios, computers, computer programs, eyeglasses, cars, planes, airports, buildings, roads, and on ad nauseam, are ALL our constructions. Nature didn't produce any of them. We did. What does it MEAN to speak of a "PHYSICAL" circle? A circle is OUR IDEA of a plane locus...
Most of our modern conceptions of math defined in terms of a universe in which there are only three dimensions. In some advanced math classes, I have learned to generalize my math skills to any number of variables- which means more dimensions. Still, let's assume that some alternate theory of the universe, such as string theory is true. Does any of our math still hold true? How would our math need to be altered to match the true physics of the universe?
Our professor today told us that the expression "7 + 5" is a single entity and a number, just like 12, and not an operation or otherwise importantly different from 12. The context was an attempt to understand Plato's aviary analogy in Theaetetus, where our professor tried to have us imagine one bird being the "7 + 5" bird and two others being the "11" and "12" birds.
This seems bizarre; while 12 is obviously the result of 7 + 5, it seems that saying they are the same is like saying a cake is the same thing as its recipe. So which is it? Is a simple mathematical equation like 7 + 5 identical to its result, or is it a different kind of thing where the similarity lies only in the numeric value the two have?
Since programming languages are supposed to be ways to express logical processes, it would seem that they would be of interest to philosophers on some level or another. For example, it would seem there are interesting relationships to be described between object-oriented programming and Plato's theory of ideas. So what are the relationships between programming on the one hand and philosophy on the other? What investigations into this area have been conducted?
Goldbach's conjecture states that every even integer greater than two can be expressed as a sum of two primes. There is no formal proof of this conjecture. However, every even integer greater than two has been shown to be a sum of two primes once we started looking. Is this acceptable justification for believing Goldbach's conjecture? Can we determine mathematical theorems based on observational evidence?
In a right angled isosceles triangle with equal sides of 1 unit and 1 unit, the third side will be sqroot(2) according to Pythagoras theorem.
But sqroot(2)= 1.414213562373095...
It is never ending.
So theoretically we cannot determine its exact length. But physically it should have a definite length! The side is touching the other two sides of the triangle, so how can the length be theoretically indeterminate but physically determinate ? Does this mean the human understanding is limited and we cannot fully understand the mind of god ? Can you resolve this dilemma ?