Always good to hear from you.
***Mathematics is logic applied to quantities. Logic is the process of creating tautologies. All tautologies are true. We can use mathematics to discover truths about quantities. The truths of mathematics apply to physics, economics, accounting, sports, and everything under the sun that can be counted. It's as simple as that.***
Let me re-word your statement so that you see where I am coming from:
Mathematics is a human standard of logic applied to human perceptions of objects. Human logic is the process of creating tautologies that satisfy human thinking. We can use mathematics to discover human perceived truths about human perceived objects. The human truths of mathematics apply in a pragmatic fashion to physics, economics, accounting, sports, and everything that humans have encountered that can be identified and counted. It's as simple as that.
***Mathematics is no more than a way of saying the same thing in different ways. For instance, 4 = 4, 4 = 2 + 2, 4 = 8 / 2, 4 = sqrt(16), 4 = ln(54.6), 4 = cos(x) + sin(y), ... there are infinite ways of saying four, and if they are mathematically correct they are absolutely true and no physicist, economist, accountant, referee, nobody at all, can prove them wrong.***
Remember, mathematics is an abstraction of what we encounter around us. For example, at one time humans probably had difficulty counting beyond what the human mind naturally processes. Instead of 10 warriors coming our way, our brains recognized a 'fair amount that we defend ourselves'. Instead of 50 warriors coming our way, our brains recognized a 'large number and we better run'. And, so on. By and by, humans began to label our sense of object, and then mathematics was born. We could then start processing these abstractions by adding numbers, etc, and finally we eventually stopped comparing our abstractions to our visual interactions with the physical world altogether. Now, mathematicians don't even bother trying to visualize their abstractions, they simply work based on rules of inference, etc, and go from there. The problem with assuming that such mathematical methods are absolutely correct and no physicist, economist, etc, can prove them wrong is that of course we cannot prove them wrong since all these abstractions are already based on the things that we encounter in the world. If our abstractions were wrong, then so would be our perceptions of objects that those abstractions are based on. However, our perceptions of objects is based on billions of years of evolution where we got the chance to develop a 'logic' that obviously is good enough for us to accept as real.
However, in spite of our senses, we can still be fooled. Quantum mechanics fooled even people like Einstein because the 'logic' that resulted from our methods was so unlike anything that we have ever experienced, that we were forced either to disacknowledge our abstractions and observations, or accept that the world looks a lot different the further away from our direct experiences. The reason we have to keep changing our theories (abstractions compared to observed reality) is because they keep coming up wrong after we observe more and more of reality and we start abstracting in more and more situations (e.g., near the speed of light, close to a blackhole, quantum sized objects, etc). Hence, abstractions of the here and now continue to serve us to address these distant lands from our immediate senses, but clearly we are constantly having to re-address our old theories to better adjust them for whatever we have yet to encounter in a remote fashion. As such, we should look at our abstractions as a tool and see any theoretical restriction on reality (e.g., reality is 100% governed by logic) as a myth. A myth can be true in particular applications (e.g., that object is a chair), but fail in other applications (e.g., the chair is composed of many atoms and free electrons that constantly become the chair and are no longer part of the chair).
***If you say you can't use math to discover truths about reality, you must be able to conceive of a situation where reality confounds a mathematical prediction. I'm curious as to what you think.***
I have more options than this. I can look at mathematics and science as being able to provide a useful tool such that I can comprehend reality with these tools. For example, I have the ability to identify objects such as chairs (thanks to clever evolutionary processes), and for that reason I can easily identify a chair (although an amoeba probably has no inkling that a chair is a valid object). The tools of human perception allow me to interpret reality in this comprehensible fashion. Likewise, if I extend my human perceptions on an abstract level, lo and behold I can extend my comprehension far beyond my most immediate surroundings. Now, I can identify quarks, supernovas, big bangs, inflationary universes, etc, and all those terms eventually translate into something meaningful to me and my most immediate surroundings. For most people they don't find it meaningful to say our universe is 13.7 billion years old, but many of us find some daily meaning in that concept. Afterall, we know what a day is, so 13.7 billion sounds like a lot of time (although we don't fully comprehend it).
If, at any time, my theory fails to explain something that my abstractions of perceptual objects are telling me, rather than change my abstractions to match the theory (which some people do with pseudo-sciences eg, astrology), the more scientific-minded people will accept that the theory needs to be changed. This is why theories are always changing, so that we don't have to change our abstractions.
Does reality conform to logic and mathematics? Well, we continue to press on 'as if' it does, and as I said in my post, this is a requirement to find meaning in the world. For example, we can't very well go through life thinking that there are no chairs can we? So we must accept as true our perceptions and the abstractions that result from them. However, in the process of doing so, we are generating myths. To question the most modern myths (e.g., modern scientific theories) can get you in trouble with the more fundamentalist-type scientists among us, but like that or not, theories change (e.g., big bang to inflationary universe), and only then does it become acceptable to say that our former cosmology is a myth (of course, don't say it too loudly unless it is an old myth - e.g., pre-1700's since modern science doesn't want to be associated with myth making).