# Are there any system(s) of mathematics whose relationship between variables bears difference to that found within mainstream mathematics?

I have been reading up on boolean algebra quite recently, for those not familiar, this type of mathematical system has much to do with the way logic is represented (and is primarily applied to, though not limited to circuits). I have learned that the result of an equation, even something as simple as 1+1, is different compared to traditional mathematics. In Boolean logic, 1+1 = True, obviously in traditional maths it's 2.

Though what I really find interesting is not the differences which boolean algebra and traditional algebra share, but their similarities.

For instance, I have found that the relationship between the two algebraic systems are not just similar, but identical, even though the product is different. This becomes clear when we replace the numbers themselves with variables that reperesent these numbers, but simutanously shines light on their relationship as well. So, AB = BA, A+B = B+A. A(B+C) = AB+AC, and so on. This is true in both systems, even though the product is very different. For those familiar with boolean algebra, you will know that the only two possible answers are 0 and 1.

So then, it seems that the results can differ, yet the relationship within a system does not, though is this actually true? Is there such a system where A+B = AB + AC, or perhaps AC = A^2+C. Don't get me wrong, this seems absurd, and would perhaps be at the boundaries (or perhaps beyond) our intellectual limits, but that's the point, such a system would be very unintuitive, but would be otherwise correct, that is if it actually represented certain properties of the universe.

• You might be interested in universal algebra, or at least some of the main branches, particularly ring theory. Studying different laws like this is very much mainstream mathematics, at least within the research community. – Slade Apr 28 '15 at 1:45
• Ah, that seems like quite an appropriate title. It really is fascinating to ponder what this subject could entail, since it seems to be diverging away from the study of numbers to something 'bigger' than said numbers themselves. I'll certainly take a look, thanks for the suggestion :) – Jim Jam Apr 28 '15 at 1:49
• @JimJam: Are you familiar with abstract algebra? It's not appropriate as an answer to your question: I would say rather than exploring possible rules, it elucidates the "ordinary rules" and explores possible structures which obey them, in total or in part. I ask because your comment feels a bit closer to that side of things, and I'm not sure how you're thinking about things. (Also, practically speaking, trying to attempt a study of UA without a solid AA background sounds like a really rough time.) – Eric Stucky Apr 28 '15 at 2:00

Certainly such things are possible. The question is: are they interesting?

For a while it can be fun to play the "symbols game", where you invent meaningless objects and play around with them. I did a fair bit of this a few years ago, and I don't regret the effort. But if you ever want anyone else to be excited about what you're excited about, then you will have to either (a) come to some objectively interesting conclusions (by which I mean, having more merit than simply unintuitive consistency), or, more likely, (b) relate your symbols to other symbols that other people care about for historical reasons.

I'm not aware of any serious attempt to relate these sorts of ideas to either the "mainstream" mathematical or the known physical universes, although as Slade mentions, universal algebra is probably the closest thing around.

For a (very rudimentary) starting point at playing the symbols game, I'd recommend the book Negative math, which is an attempt to take seriously that age-old middle-school question "Shouldn't minus times minus equal minus?". IMO, the book thinks a little too much of itself, but if you can get over a bit of pretension then you probably would enjoy the ride.

• Hey Eric, thank you very much for taking the time to respond, am sorry that it took me so long to (just came back from HK! :D). Anyway, I think I see what you mean. For patterns within maths and science to have any merit it should either A) Already bare significance by building off other, well know systems, the most recent probably being calculus, or B) Hold significance independent of itself from any other system. – Jim Jam May 9 '15 at 0:47
• Also i'll take a look at that book as well as abstract/universal algebra, I can probably read the book at the same time I look into these new found subjects:) – Jim Jam May 9 '15 at 0:49
• @JimJam: If you think that calculus is the most recent well-known system in math, you aren't going to need to venture outside the realms of mainstream mathematics to discover some incredible mathematical worlds. Best of luck on your explorations :D – Eric Stucky May 9 '15 at 5:32
• (Protip: it's not easy to learn algebra in an unstructured way. I would recommend getting your hands on a solid algebra textbook: see this question as well as a solid linear algebra textbook. Linear algebra is not very close at all to what you want, but it might help you get some intuition for abstract algebra) – Eric Stucky May 9 '15 at 5:38
• Actually, I'd really like to talk to you on chat sometime. Go on the main room and say something; this will automatically create your chat.stackexchange profile. Then let me know in this comment chain, so that I can invite you to a semi-private room and we can try to make schedules work – Eric Stucky May 9 '15 at 5:41