The Laws of Mathematics are Unchangeable

Those are some interesting arguments. A note on base-systems (for readers): Base systems are basically how we count. Our current system is base on tens, hence the name 'ten-based'. This is so because we have 10 digits "0,1,2,3,4,5,6,7,8,9". The digit in the 'ones' place tells us how many single units there are, the 'tens' place how many tens etc. So 23 tells us we have 3 ones and 2 tens. Now, what if we switched to the two-base system called binary? To start, we only have two digits: 0 and 1. The first digit still tells us how many ones we have, but the second tells us how many twos we have, the third tells us fours, the fourth tells us eight etc. So in binary, 1 is what we know as 1, but 10 is what we know as 2 (we have zero 1s and one 2). 11 is three, 100 is 4, 101 is 5, 110 is 6, and so on. Ternary is three-based, with the digits 0,1, and 2. The first digit tells us how many ones, the second tells us how many threes, the third how many nines. 1 is still one, 2 is still 2 (we can use 2 now, remember?), but 10 is 3 and 11 is four. "All right, the main argument that I have to put forth here ..." What my opponent does here is postulate the existence of mathematical laws. He states that the laws of mathematics exist, and we are simply discovering them. How do we know such a set of laws exist? Where is the evidence for this? If such a system were to exist, would it be a set of laws that encompass all branches past, present and future of math? Such a system which encompasses all of math is impossible, mathematical ideas such as the Continuum Hypothesis (detailed in the article) cannot be proven or disproved (it has been proven that the Continuum Hypothesis cannot be proved; it has also been proved that it cannot be disproved). In this case, there is a property of infinity which cannot be described by this 'complete set of laws' which we are simply discovering. The simple fact that there exist such areas which can neither be proved nor disproved casts doubt upon the existence of such a set. I would contend that math is something that we have defined as humans. Math is a collection of principles, symbols and ideas that have been codified. From basic addition to number theory (my favorite), we have 'invented', so to speak, that which we call math. It is useful in describing the world around us simply because we wish it to be so (in the basic sense that we define variables and symbols to real world calculations, making them easier). If we want to know the length of a leaf (a long time ago), we invented symbols that described it length. In this same way, language is something we have invented to describe the world around us. Now, about your infinity..." Guesses far from describe what we currently know about infinity. The examples I cited have been rigorously proven by various mathematicians. The argument I am making here is not that principles change from saying " 'A' is true" to " 'B' is true", but from " 'A' is true, and some other statement 'C' is true too". That is, we are constantly adding to our understanding of math. "Now, on to your argument about changing from a ten-based system..." What I am saying here is that you cannot restrict mathematical systems by saying that you cannot change aspects of it. Depending on the system that you are in, the laws of mathematics change. Math operates in many different systems. Mathematical laws are different in binary and ternary. These are different systems that we can use in calculations. As they are 'modes' of math, why is it that they cannot be used? The only reason provided by mongeese that different base systems cannot be used is that it "clashes with the clause [he] included with [his] argument in Round 1." There is no reason presented here that we cannot use different base systems. Base systems are perfectly valid examples of changing mathematical laws. Changing bases are an integral part of math, and discarding them from this argument is similar to discarding addition and the likes. "And what we assume to be a law..." Again, I apologize if I was not clear here. What I am saying is that we, as humans, have laid down the basic laws of math. The most basic principles are ones that we have set down which cannot be proved. Mathematics is simply a construction of man. As we have invented language to help us solve our problems and articulate ideas. Perhaps assumption wasn't the best word to choose, the phrase 'Math, truly, has no foundation. It is based on the rules we have laid down at its most basic level'. Because math is a creation of man, and therefore its laws can change, also because changes in base change mathematical laws, the laws of mathematics are changeable and the resolution is negated.