I am offering an essay that brings Objectivist epistemology to bear on mathematics in a novel way, leading to new conclusions. Its title is "Understanding Imaginaries Through Hidden Numbers." It is available here:

The article's title refers, among other things, to the imaginary unit i used in complex numbers, but all the more basic ideas of math are covered as well. Most importantly for Objectivists:

o How do we arrive at concepts of numbers?

o Why do we learn mathematical ideas at roughly the same time as the rest of our early concepts?

o How would one define "number"?

o What is the meaning in physical reality of the so-called imaginary/complex numbers?

First, what is a "hidden number"? In a mathematical theory developed according to contemporary standards of rigor a mathematical theory will set forth its undefined terms and the postulates which give the undefined terms their semantics. Nothing is "hidden". The method of development is transparent, straightforward and logical. This is how the theory of (so-called) real numbers is presented nowadays. Likewise for complex numbers. A complex number is an element of the extension field over the real numbers generated by the irreducible polynomial equation x^2 + 1 = 0. The root of this equation is the (so-called) imaginary unit, usually designated by the letter i.

Second, we arrive the concept of numbers by counting. Kids can do it by the age of two usually.

Third, we learn mathematical ideas the same way we learn the names of things, the names of actions, the meaning of relations etc.. Kids can do this by the age of three usually. There is no magic here.

Fourth. How would one define number. What kind of number? Integer, rational, real, complex, quaternion, octonian? Which? Please be specific. Use standard mathematical terms if you are going to discuss mathematics.

Fifth, what is the physical meaning of the so-called complex numbers. The usual physical interpretation is that of angular phase. Complex numbers are given by an angle and a magnitude (relative to some frame of reference). This is very nifty for discussing harmonic oscillators and is this interpretation is even more nifty in formulating the concept of a quantum state. Quantum states are unit vectors in a Hilbert Space. The set of operators with quantum significance on this space are the so-called Hermite Operators.

Now let me ask you something. Has your magnum opus added one substantial theorem to the theory of analysis of complex valued functions that is not already in the literature?

If not, what is it you are bringing to the table?

Ba'al Chatzaf

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## BaalChatzaf

First, what is a "hidden number"? In a mathematical theory developed according to contemporary standards of rigor a mathematical theory will set forth its undefined terms and the postulates which give the undefined terms their semantics. Nothing is "hidden". The method of development is transparent, straightforward and logical. This is how the theory of (so-called) real numbers is presented nowadays. Likewise for complex numbers. A complex number is an element of the extension field over the real numbers generated by the irreducible polynomial equation x^2 + 1 = 0. The root of this equation is the (so-called) imaginary unit, usually designated by the letter i.

Second, we arrive the concept of numbers by counting. Kids can do it by the age of two usually.

Third, we learn mathematical ideas the same way we learn the names of things, the names of actions, the meaning of relations etc.. Kids can do this by the age of three usually. There is no magic here.

Fourth. How would one define number. What kind of number? Integer, rational, real, complex, quaternion, octonian? Which? Please be specific. Use standard mathematical terms if you are going to discuss mathematics.

Fifth, what is the physical meaning of the so-called complex numbers. The usual physical interpretation is that of angular phase. Complex numbers are given by an angle and a magnitude (relative to some frame of reference). This is very nifty for discussing harmonic oscillators and is this interpretation is even more nifty in formulating the concept of a quantum state. Quantum states are unit vectors in a Hilbert Space. The set of operators with quantum significance on this space are the so-called Hermite Operators.

Now let me ask you something. Has your magnum opus added one substantial theorem to the theory of analysis of complex valued functions that is not already in the literature?

If not, what is it you are bringing to the table?

Ba'al Chatzaf

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