Proofs of existence, diversity as value system, math(s) as a human construct.
Concerning things and properties.
The very notion of a thing without properties (again, in the broadest sense of the word) cannot be conceived, for the moment one can conceive of a thing (unicorn) one can begin identifying properties which mark it as distinct from other things. I might put to you that there is only one thing that does not exist: non-existence. There are no items in this category. It is the empty set.
And so, I do not believe it is possible to define existence through a thing which exists and an identical thing which does not, as the latter is an impossibility. By virtue of being identical, it must exist. Or, if it does not, the only comparison which can be made is within the realm of existence vs. not, which means it cannot be an identical thing.
And so, I maintain my stance (and Descartes) that 'I exist' is a better foundation than things may exist, since it validates the latter. As well, it is simply the most self-evident statement I can think of, for each person that states it. I do not think one could get very far on the first principle that 'things may not exist,' so its implied counterpart is of no greater value, as they say exactly the same thing.
This gets on (in a slightly meandering way) to what I believe is the most important aspect of life itself, and that which I think we as a race are taking the most devastating toll on: diversity. The very definitions of everything we know stem from all the ways that a thing differs from all other things (all properties shared by a type can conceivably define the type, in a rough-woven scenario, and differences within the type can specify further, eventually unto uniqueness), rather than from non-existence. In this sense, I believe all of existence can be pared down by the idea of contrast. The only way you can really damage the universe as a whole is to remove parts of it such that there are fewer things by which to distinguish and define the rest of the things. This is especially true in the case of Life Itself, where we attempt to classify and document every living creature by its attributes, and can only do so when there are all the other creatures flitting about with which to compare common or divergent attributes. Destroying millions of species (without paving the way for others evolving to take their place) is a crime against life itself, and that is far worse than any crimes against humanity, to my mind. To think that humanity is the most powerful force known in opposition to life itself, even as we have the greatest potential of all living things, is somewhat appalling. Funny ol' world.
It is this notion of diversity inherently giving value to all that exists which sparked one of my favourite ideas, and if we do not start doing something like this soon, I may just write it out as fiction at some point. That is, I would like for us, before we lose the capacity, to bombard the universe with capsules teeming with the ingredients for life. Perhaps some billions of bacteria in some special goo, or something, to give Life Itself the chance at gaining a foothold on other planets. Not for our use, but just because it is great.
On to maths!
I must chuckle at this a moment, both in appreciation of the role reversal, and at your bringing up of the case of using maths to accurately explain empirical phenomena. The latter because I am making nearly the same point, from a reversed perspective again: maths is used to accurately explain empirical phenomena because that's what we've created it to do! In its simplest form, I take this thing, that thing, and that other thing, and I create a novel(TM) construct when I tell you there are THREE things. As far as a mindless universe is concerned, there is this, that, and the other, and it takes a mind to truly quantify. Most of the rest of mathematics, I believe, can be built upon the idea of quantification. Quantifying is a mental construct which allows the mind to create a new plane of abstraction in which things of a type, or sharing some property(ies), may be assigned a specific and unique value based on the number of individual instances at hand.
Once ideas of quantity as explicit value begin to concretise in the mind, the next plane of abstraction becomes available, allow us to play with quantities alone and forget what the items are: for example, we can combine 2 of this and 1 of that, and observe that we have 3 items. From this we generate a rule which we claim is forevermore true: that 2 and 1 is 3, and we can add a label to help us concretise this new level of abstraction by calling it addition. The rules are generated by the mind based on what's observed. That some may apply universally (maybe) to novel scenarios does not negate their origin in the mind, rather than Out There.
I could claim that 2+2=5, and I think you'd have a deuce of a time proving me wrong. You can say 2+2=4 all day, but it doesn't actually disprove my position. However, mine isn't a great one for dealing with the world, so I will abandon it after this conversation has moved on. I created a mental construct of equal value, theoretically, to the regularly held viewpoint.
Now, having said all that, I'd like to come back to, and vehemently agree with, your observation that maths actually works in elegantly and simply describing aspects of things which seem initially to be extraordinarily complex. However, I would place the laud for this not at the feet of the mythical magic of mathematics, but in the very real and INCREDIBLE capacity of the human brain to have created this most precise and exquisite of tools to help us interpret and develop within our universe. I do not think almost anyone realises how beautifully a brain works to be able to do this, and so much more on top of it all.
I might go so far as to say that our brains and minds are the most underappreciated things on the planet.