• João Neto: Do you think math concepts exist independently of persons (aka, mathematical platonism)?
• Alexander Nikitin: Define 'exist'. --- 
• JN: Well, I overload the word 'exist' with two meanings: (i) existence independently of persons (let's call it exist-1) and, (ii) existence due to persons (exist-2). It's the separation of Maps and the Territory. 
• AN: You can argue that only mind exist. Everything else are just ideas in your mind. Then you have an idea of natural numbers in your head. And you have an idea of a set of axioms and deduction rules that define your space of natural numbers. Then you start doing different things with your natural numbers following the rules and you find that the space has a structure and properties that depend on those axioms and rules only and nothing else in your mind. Your mind can do whatever it wants. It can have any other ideas, feelings and states, but whenever it follows the axioms and rules it always gets the same results. Does that mean that there is something that exists independently of your mind? 
• JN:  But there is evidence against solipsism: we have lots of data from our senses and tech extensions. And my mind cannot do whatever it wants, especially when I'm driving :-) We are more than one human, why would one single mind be special? And if we assume two or more minds (a belief that literally every one follows) then how could we explain the coincidence of sense data? The belief that 'only mind exist' does not help us at all, it dissolves everything and give us nothing. 
• AN: If you believe that the 'real' world exists and math is just an artifact of human brain activity. Then notice several things: (1) You can define a mathematical system as purely abstract and symbolical without any attempts to model anything 'real'. You can even ask a computer to generate a random set of symbols, axioms and rules for you to exclude any 'subconscious' mapping to anything real; (2) You can give that math system to any other people and/or computers and notice that if they follow the rules they always get the same results regardless of the properties of their minds; (3) If you eventually manage to map your system to something 'real' you will notice another strange thing: the results of physical experiments will always follow predictions derived from the system once the mapping has been established, but it NEVER work in the opposite direction - physical results NEVER disprove any conclusions derived from the axioms and rules of the math system. They can prove that the mapping was wrong, but they can't 'bend' the 'truths' of the math system. As we see math is self-sufficient. Substrate independent. And physics always follows math. Then what is more 'real'? 
• JN: The first and second point also work for board games. And games can be formalized as math objects. However, defending that, say, Chess exist-0 is a very strange assertion for me. I don't see the relevance of the third point. Some math mappings are adequate to formalize scientific models (since there are infinite mathematical models, only a vanishing part of them are really useful). And, sure, enough counter-evidence can cancel the previous adequacy between model and data. So what? Why would we want to use data to bend Math? We just try to find or create another Math model that does the trick. They are tools just like anything else. We use many maps/models. We could imagine a spectrum from totally objective to totally subjective. Humans developed a discipline to deal with those at the objective extreme of that spectrum and called it Math (Physics, which is just another family of models, is a close neighbor). 
• AN: The third point is important because it tells us that math is not just a map. It has predictive power. A map doesn't give you more information than you already know from your 'real' data. It shows you only that part of the territory which you already have experienced. Once you get mathematical model you instantly know everything about your territory. All human engineering is based on this idea. Every day we build bridges, planes, skyscrapers - 'real' objects that never existed before - and we can do it successfully because we rely on the empirical fact that once we get the math right we can be sure that the 'real' system's behaviour will follow the math. Our experience tells us that the 'real' world is not random. It follows certain rules. That means that the rules exist. The rules are abstract concepts they are not a part of the physical world because they define it. So if you believe that the world is not random then you should accept that abstract concepts can exist independently of anything in the 'real' world. I asked you to define 'exist' in the beginning because abstract concepts don't exist in the same sense as 'real' things. We can not put them in any specific place and point in time. They exist outside of space-time. I think I was not clear from the beginning. My view is not just that abstract mathematical objects exist. My belief is that mathematics is the only thing that exists. Our 'real' world is just one of all possible mathematical systems that has 'self-aware' objects that precept their environment as 'real'. That's all. 
• JN: When you say that the fundamental basis is Math, the only thing that exists, and outside space-time, you are putting yourself into a position that cannot be settled by evidence. Which is not unreasonable since we are talking metaphysics. I recently already read similar arguments from people like Bill Taylor, Massimo Pigliucci or Steve Landsburg (all quite clever chaps). I think we cannot possible breach the abyss between the Territory and the Map (for me, we are entirely Map denizens). I just don't like positions which are 100% argument and 0% evidence. So, I try to minimize my own 100% argument beliefs: I don't assume anything from the Territory except that it generates events. And that's because we are able to measure those events (these are partial measures due to the limitations of our sense apparatus). These events, as you said, do have some regularities that we adapt ourselves to them by customs and culture, and formalize some into scientific laws. Why there are regularities? I don't know. Nobody knows (anyway, as far as we know people would be impossible in a more random universe). It's a bit like Hume's guillotine for ethics, there's an ontological guillotine between existence-0 and existence-1. I would bet that no one will ever cross it (in fact, from my position, that does not even make sense). Now, when you say: "Our experience tells us that the 'real' world is not random. It follows certain rules. That means that the rules exist." I think you are falling into Whitehead's Fallacy of Misplaced Concreteness (aka as ET Jayne's Mind Projection Fallacy). Rules imho only exist-1 in our minds. Perhaps using a less objective example, I can explain it better: humans share lots of cognitive bias. We can extract rules from well-made psychological studies. But we don't assume that these rules exist outside its scientific context. For me, the same happens with electrons, QM, etc. I see some wisdom in the "Shut Up and Calculate" attitude. In this case, we can only appreciate each other arguments. At the end of the day, we are confronting aesthetic positions not truly empirical ones.