What I meant by adding psychology into the mix was that "seeing math everywhere" requires an attention to what is sensed and in stage two, what is perceived. A focus on doing this, perceiving, seems to be more prominent in certain disciplines. Some math folks with math walks stress it, but many do not as pointed out by jgill. In the arts and in art museums, perception and paying attention to one's surroundings is central. This stress is what I am promoting in understanding mathematics and computing. I am not suggesting everyone do it, but it would seem to correlate to an "arts way" of interpreting.

Good quote for Putnam and for the focus on pattern-seeking. In my classes over the past 5 years, students have been encouraged to interpret things they see on a daily basis as mathematics, or its natural abstract extensions (statistics, computer science). In self-critique, I wonder where this interest originates. It might be based on an artist-workflow where perception is strongly valued and taught in classrooms. So, enter psychology.

I think you are right. Thanks for asking them.

I appreciate all of these excellent points and suggestions. Let me clarify my interest in the posed question. The "mathematics is everywhere" way of seeing is targeted to mathematics education,
but can easily be extended to abstract mathematical areas of statistics and computer science.
I see it as a means for teaching and learning. I am not suggesting that mathematics is the main,
or only, way of seeing since one can walk around the house or apartment and "see" objects from multiple viewpoints, and not just mathematics. For example, chemistry, physics, design, social issues (e.g "Ways of Seeing" by Berger in art criticism) are all worthy ways of seeing.

Perhaps a more accurate question on my part would be "What are the philosophies of mathematics that underlie the movements in math education based on math trails/walks?"

Some potentials are included in this thread. Tegmark in "Our Mathematical Universe" takes the ontological view that "the universe IS mathematics". As much was said by Galileo and others. And I agree that such views and discourse do contribute an answer to "mathematics everywhere." Some other movements such as digital physics and Wolfram's "new kind of science" seem similar to Tegmark's thesis.

I may be seeking a simple answer to a complicated question. There may be no single philosophy of mathematics that is situated empirically in seeing math in everything. I am not seeing anything that stands out here: https://plato.stanford.edu/entries/philosophy-mathematics/

Phenomenology and Empiricism might also contribute as philosophies as well as embodied mind theories.