>No. Formal thinking involves more than just logic, and I get the impression that by "formal logic" you mean symbolic logic.
Well, I was guessing at what you mean, but I'm also too glib - I should have said "formal system" - a formal logic plus additional primitives and axioms as needed. Or, more broadly, as wikipedia puts it: A formal system is, broadly defined as any well-defined system of abstract thought based on the model of mathematics.
>Mathematics is a product of the mind, not the world.
I've said a few times now that historically, formal considerations led to the rules around negative numbers and so-called imaginary numbers. That's giving your point a lot of acknowledgement -- as it justly deserves. But, what you are not acknowledging is that those same rules, used as they were without sensible mappings and conceptual underpinnings, led to a protracted period of confusion even among careful thinkers and mathematicians. Read the book, or some of the condensed histories I've referenced. So, the system we teach today is not without difficulties, that are not resolved simply by claiming they are the result of pure reason, formal reasoning, or whatever you want to call it. In fact, other systems are even possible.
1. There is evidenced by the historical record, a continued desire, among general population and mathematicians themselves, for "models" to map the new abstract operation to. So they were developed. Let's use them and not make everyone go through the same old confusions. But Peter has hit on one of the nagging lingerers: (-) x (-) = (+). Just look around the web, maybe starting here:
There is no one compelling paradigmatic mapping, though things involving money tend to stand out a bit.
2. Part of eliminating confusion might be to eliminate certain old terms that I'm sure are still used everyday- like "negative quantities". Negative number is better, alongside an explanation that "number" is not just about "quantity" anymore. The sign is a new and separable component. If it maps to anything its not to quantity.
3. Now, I admit that its more problematic to replace the terminology for "<" & ">", but really, the old terminology is not good in the integer setting.
R.H. says: >I don't care how vivid an example of a mathematical concept might be, the example itself is not mathematics.
Now I think you are just drawing boundaries or arguing semantics. I don't care much about that. To me, math is what history shows us is involved in the enterprise of mathematics, which includes a lot more than what you want to consider. My bias against your view is that it seems to be a particular part of that historic enterprise, that wants to paint a religious picture of mathematics as being pure and transcendent. I'm not buying it these days.
The particular notion of "mindfulness" (your term) you wish to enshrine as leading unerringly to "the truth" is utter, ... you know. Very mindful people found the (historic) lack of mappings, and non-sensical terminology and conceptual foundations (e.g., calling a signed number a "negative quantity", and "less than nothing") to be serious flaws.