The Laws of Mathematics are Unchangeable
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after 11 votes the winner is...
ToastOfDestiny
Voting Style:  Open  Point System:  7 Point  
Started:  4/12/2009  Category:  Science  
Updated:  8 years ago  Status:  Post Voting Period  
Viewed:  9,539 times  Debate No:  7819 
Debate Rounds (3)
Comments (32)
Votes (11)
I'll start with a definition:
Mathematics: the science of numbers and their operations, interrelations, combinations, generalizations, and abstractions and of space configurations and their structure, measurement, transformations, and generalizations http://www.merriamwebster.com... My argument is that as long as the symbols and functions of math do not have their meanings changes, the laws of mathematics are unchangeable, no matter what you try to prove.
I would like to thank mongeese for this challenge and look forward to an interesting debate. A quick opening: as long a new symbol or concept is added to "math", we have a change in the laws in terms of the addition of new laws. That is, the moment a new branch of mathematics is developed or an old branch is modified, the laws change. For example, let us examine the United States' Constitution. Every time an amendment is added, the Constitution changes. Therefore, the Constitution is not unchangeable. Therefore, Con should immediately win this debate. As you will see further on in my case, math is dynamic and constantly changing. I will take the resolution as: "Resolved: The laws of mathematics are unchangeable" Definitions: Mathematics  accepted (with the note that applied math is mathematics  physics for example is applied math) Unchangeable  not changing or to be changed http://www.merriamwebster.com... Change  to make different in some particular http://www.merriamwebster.com... Laws of Mathematics  basic principles upon which mathematics is founded (my own definition, I hope it is rigorous) Basically, I will prove that the laws of mathematics have been changed (ergo, are changeable) and can still be changed (again). First, we must look at our definition of unchangeable. For something to be unchangeable, it must not be altered or able to be altered (not to be changed). The laws of mathematics are always changing. The history of mathematics shows us that concepts are always changing. To list a few areas where the laws have changed: 1) Zero; introduction of, use of 2) Calculus; introduction of, definition of, specifics of 3) Limits; definition of 4) Infinity; rules defining 5) Space; Euclidean and NonEuclidean I will touch on infinity in this round. Contention One: Infinity is an example of changing mathematical laws Sources http://www.mathacademy.com... Infinity began as a concept of the unending. Aristotle laid down basic ideas of infinity. He stated that actual infinities could never exist, which we now know to be false, infinities do exist within mathematics. Here we have one example of a change in mathematical laws (the resolution now goes Con). Nevertheless, I will continue. When Newton and Leibnitz developed calculus, the need for infinity became apparent. They dropped the issue, believing it to be impossible, but the laws of calculus depend on infinity. Side Note: The development of calculus, and its laws, gives an example of how the laws of mathematics are changeable. Calculus added laws to math, therefore a change. Back to infinity. Georg Cantor finally laid down some laws defining infinity, which is called Cantor's Set Theory. My source provides a good explanation of this, so I will not explain it here. Thanks to Cantor's work, we have defined several aspects of infinity. Infinite sets can have parts removed from them without getting any smaller. Another important result of his work is the proof that infinities can be larger than each other. They are denoted by the Hebrew letter Aleph, with Alephnull being the smallest infinity, Alephone being larger, Alephtwo larger etc. Here we see a drastic change in the laws of mathematics. Previously, the concept of infinity was completely undefined and thought to be nonexistent. With the advent of Georg Cantor, the Laws of Mathematics had to be modified to fit these new findings. My opponent's argument: "My argument is that as long as the symbols and functions of math do not have their meanings changes, the laws of mathematics are unchangeable, no matter what you try to prove." What mongeese is saying is that as long as symbols and functions are not redefined, the laws of math do not change. Here are a few points to think about. Math is constantly evolving, with new fields emerging and old fields being redefined. There is still much work to be done concerning infinity, and even more in other fields. Also, mongeese's argument is inherently flawed. It assumes that math is unchanging and concrete. Math is not like this, however. It is by definition constantly changing. Like a set of laws that a country lays down, what may be permissible at one moment may not be allowed the next. This whole argument is based on a false assumption. Here's an interesting thing to note, the phrase "no matter what you try to prove". What if I were to prove something radical, like "2x2 is not equal to 4" (coincidentally this can be proved in a system that is not tenbased. In ternary, for example, the correct notation is 2x2=11)? If I prove something which is not included in the laws of mathematics, they change to accept this new fact. When it was proved that there are more real numbers than rationals, the laws of mathematics changed to accompany this. When infinities were accepted as existent, math modified. And a final note. Technically, everything which I have proved above can be considered false. In pure mathematics, statements like 2x2=5 can be proved to be true. The only reason we accept that 2x2=4 is true is because we have assumed the basic properties of math. Math, truly, has no foundation. It is based on assumption at its most basic level. To explore this more: http://en.wikipedia.org.... On the most basic level, all that is necessary for a mathematical change is a change in assumption. I think mongeese was trying to block this, but saying that you cannot change symbols and functions is not allowing math to be math. Quite simply, math changes, the laws change, and the resolution is negated. 

Thank you for responding, ToastOfDestiny.
All right, the main argument that I have to put forth here is, just because we think that a statement is a law of mathematics, doesn't mean that it actually is. http://en.wikipedia.org... (Click on abstraction (mathematics), pages with () don't hyperlink properly.) Zero existed before we thought of it. Before Calculus was started in 1820 BC, the natural log of "n" was still equal to the area of the shape formed by x=n, x=1, y=0, and y=1/x; we just didn't know it yet. When we write "new" laws of mathematics, we aren't changing them; we are discovering them. For example, before Newton wrote his laws of physics, they affected all objects. Now, about your infinity. Just because we made some guesses about infinity, doesn't mean that they were laws of mathematics. Maybe what we currently think to be the laws of infinity may not be the actual laws of infinity, but that doesn't mean that there are no laws of infinity out there. Any law of mathematics that has been proven to be wrong never was a law of mathematics in the first place. Now, on to your argument about changing from a tenbased system to a threebased system. In this case, the symbol 11 no longer stands for eleven; it stands for four. Changing what a symbol represents clashes with the clause I included with my argument in Round 1. "Math, truly, has no foundation. It is based on assumption at its most basic level." And what we assume to be a law of math may not be a law of math. It is an assumption of something that may or may not be true. And 2x2=/=5. Simply, what we think of math changes, as we discover more and more things, but true mathematics never changes, and the resolution is affirmed.
Those are some interesting arguments. A note on basesystems (for readers): Base systems are basically how we count. Our current system is base on tens, hence the name 'tenbased'. 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 twobase 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 threebased, 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 tenbased 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. 

Thank you for your little math lesson.
"How do we know such a set of laws exist? Where is the evidence for this?" Okay, mathematics, like physics, is a branch of science. The laws of physics have existed since the dawn of time, and the laws of physics aren't written down anywhere. We are slowly discovering all of these laws of physics. We humans don't have the power to change the laws of physics. These things that apply to physics apply to mathematics as well. At the dawn of time, if you wrote the equation x�x6=0, then used the quadratic formula, which is: x=(b�√(b�4ac)/2a, then you would get the two solutions to the equation, x=3 and x=2. http://en.wikipedia.org... However, the quadratic formula was first discovered in 628 BC http://en.wikipedia.org.... I say discovered instead of invented because it still existed, but we just didn't know that it worked, refuting your point. It existed long before we thought of it and wrote it down. We didn't invent math; we just discovered that it worked and applied it. The same applies for physics, when long before Newton was born, if you pushed an object into frictionless space, it would continue to move without acceleration until an unbalanced force acted upon it. "Guesses far from describe what we currently know about infinity. The examples I cited have been rigorously proven by various mathematicians." Anything that a mathematician can prove without a doubt is a law of mathematics. If a proof is disproven, then it obviously wasn't proven without a doubt. Now, about what Aristotle said, "Since no sensible magnitude is infinite, it is impossible to exceed every assigned magnitude; for if it were possible there would be something bigger than the heavens." Aristotle was attempting to apply mathematics, by comparing the infinite to the heavens. Applied math, such as physics, is different from regular math, as you already said, "With the note that applied math is mathematics  physics for example is applied math." Just because we make different guesses about infinity, doesn't mean that infinity constantly changes. The concept is just sitting out there, laughing at our mistakes, waiting for us to finally discover and prove what it really is. (Note: That sentence there was metaphorical, not literal.) "What I am saying here is that you cannot restrict mathematical systems by saying that you cannot change aspects of it." My argument against that is that changing the base number of the system doesn't change the actual values of the number. In our tenbased system, 7 is supposed to represent seven, but in the binary system, 111 is used to represent seven. The laws of mathematics do not use symbols to represent values; they use values. The symbols were come up with by humans. The values were not. (Note: In my explanations, numerals are symbols, while words are values.) It is common knowledge that two times two is four. We commonly express this as 2x2=4. If you were to use a binary system, this would be expressed as 10x10=100 (which, ironically, is a real solution in either system). In both cases, two times two is four. We may read it in a different way depending on the system we're using, but in "natural" laws of math, the values are all that matter. "Changing bases are an integral part of math, and discarding them from this argument is similar to discarding addition and the likes." This is beyond the unchangeable laws of math, as this is just our interpretation. "Again, I apologize if I was not clear here." Don't worry, you were loud and clear; I just disagreed with you. "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...." No, we have not. We have discovered them. Math is both very abstract and very real. We did not decide that two plus two is four; we discovered it when we put two apples with two oranges. We did not decide that five times three is fifteen; we discovered it when we laid out five rows of three pebbles, and counted them. "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." CONCLUSION: Because math has always existed, its laws are "set in stone", and we as people can only discover new aspects of math. Change in base does not change the values of the numerals, thus not changing the law that is applied to it. The laws of mathematics are unchangeable, and the resolution is confirmed. Thank you, and I look forward to a brilliant response.
This has been a fun and interesting debate, a great break in the tedium of spring break, and I would like to thank mongeese for that. "Okay, mathematics, like physics, is a branch of science." I have two very important points to make here. First, the laws of physics can change, and second, my argument about unprovable areas in math was completely ignored. 1) Concerning the laws of physics. The laws of physics can change. Quite easily, actually. Now, we have evidence that certain particles, called 'force carriers' exist. These particles, as is evident from their names, carry the four forces in our world: the electromagnetic force, gravity (we're still searching for evidence for this force carrier), the strong nuclear force, and the weak nuclear force. Respectively these are photons, gravitons (still hypothetical), gluons, and the W and Z bosons. Here's the crazy part: at temperatures in the universe today these particles exist together. However, we have reason to believe that at the beginning of the universe the four forces were all carried by a single particle. That is, instead of these five little particles running around mediating forces, we had one doing all their jobs. Also, forces diminish in strength with distance. Imagine that there was only one gigantic planet in the universe. As you get closer, you feel its gravitational pull more. But the further you get away, the weaker the pull. In our universe with three large spatial dimensions, gravity diminishes by a rule called the inverse square: 1/r^2, where 'r' is the distance from the object. So if I am 10 meters from the object, the force felt is 1/100 as strong. However, if our universe had only two large spatial dimensions, gravity would diminish by the rule 1/r. 10 meters away, gravity would be 1/10 as strong. This is only one example (electromagnetism follows the same rule) in which the laws of physics can change. 2)Areas which can neither be proved nor disproved. Mongeese does not address my argument on areas which can neither be disproved nor proved. The Continuum Hypothesis, for example physically cannot be proved nor disproved. If a set of overarching laws existed, surely they would describe all of math. Mongeese's arguments necessitate the existence of a complete set of mathematical laws which we are moving closer to. However, such a set of laws which encompass all of math are impossible to achieve, which is demonstrated by such areas. If therefore no such set of laws exist, then math is something we are laying down for ourselves. To summarize the above two points, just because something is a science doesn't mean it has a set of alwaysexistent laws, and even in fields where such laws exist, the laws are not unchangeable. "Applied math, such as physics, is different from regular math, as you already said..." I was noting that physics is pure mathematics which we have molded to describe the real world. We prove different things about infinity, adding to the laws of mathematics. "The concept is just sitting out there, laughing at our mistakes, waiting for us to finally discover and prove what it really is" The concept is constantly growing, like a silent sapling (also metaphorical). "Math is both very abstract and very real" And yet we develop areas of math without application in the real world. Fields like number theory which do not have bases in real world problems defy this argument. In conclusion, a set of overarching mathematical laws cannot exist as there are areas which cannot be explained. Just because a study is a science does not mean it will have laws, and laws that are present can change. The laws of mathematics are changeable, and the resolution is CONfirmed (negated). 
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Congrats, mongeese. I'm still puzzled how I pulled this off.
Coincidentally, 10x10=100 and 1x1=1 are true in any base system, as 10x10 is the basenumber squared (in binary it's two squared), and 1x1 is what we know as 1x1 in any system.
2. It's not that we can't prove it in the sense we lack the knowledge  we can't prove it in the sense it is impossible to prove.
1. We don't know everything there is to know about chemistry and physics, either; for all we know, there may have been some other factor in the Big Bang that we still haven't identified, such as maybe a fifth grand force that contracted, then released, the Big Bang.
2. If you can't prove which of two men fired their arrow at a target first, that doesn't mean that neither one of them hit the target; likewise, just because we can't prove something, doesn't mean that there is a very real answer that has always existed, and we might actually never know.
I doubt man actually created 1=1, it was discovered.