I would say that “Nature” in the above claim would mean something along the lines of... The Observable Physical Natural Universe.
You really shouldn't define a word by referencing that same word (or derivations thereof).
Here you are defining "nature" with the term "natural" - and thus your definition is, alas, hollow.
Can you try again with a definition?
Uniformitarian Naturalism seems to be an assumed foundation that at least part of science is based on, and which seems to make the three Claims I stated above.
Can you provide a link that describes Uniformitarian Naturalism in more detail, please?
I have Googled, and I have found plenty with regard Uniformitarianism, some of which even puts it at odds to Naturalism (e.g.
https://www.slideshare.net/Mahesh3391/uniformitarianism-versus-naturalism).
Where, for example, have you established that the three claims in your OP are claims of Uniformitarian Naturalism?
But other theories proposed, especially those concerning the historical past, do seem to be built upon these three Claims as bedrock foundations. Even Dating Techniques rely on these three, or am I out to lunch?
I would suggest that the third claim in your OP is rather just a parsimonious assumption until other information arises that refutes it.
E.g. if you uncover a system that is behaving in a certain repeating way, it is parsimonious to assume that it did this in the past.
You then take that assumption, run with it, make models by which the assumption can be tested with other data.
If the tests pan out, great, the assumption survives; if not, revise the assumption until the model covers the new data as well.
That is how science works.
Can anyone prove these three claims or are they just assumed without proof?
In turn:
1)
"Nature is all that exists."
Well, you'd need to define what it means to exist, and this is a metaphysical issue.
Science starts from such places, but can not be used to prove the assumption correct.
Science also doesn't "prove", but merely reaches a high probability of something being consistent with reality.
Can maths, therefore, prove the claim (mathematics being where "proof" can be found)?
As much as, without further elaboration of what the terms mean, it can prove that Szark is Thumst.
2)
"Everything can, and indeed must, be explained by time plus chance plus the laws of nature working on matter."
There are many philosophers who would argue that there is zero chance in the game at all, just a lack of understanding of the precise workings of the "laws of nature", or a lack of understanding of the way the pieces are positioned, such that "chance" becomes a convenient means of forecasting possible outcomes in the absence of such knowledge.
Others would argue that chance is inherent in the structure of the universe, that even with such knowledge of the pieces and the workings, that the outcome is indeterministic.
So it depends on how one is defining "chance" in this regard: is it simply due to a lack of knowledge that could be known, or is it inherent despite knowing?
Further, if the latter, then "chance" becomes a law of nature, and thus either way the "plus chance" is simply not needed in your claim.
As to it being assumed without proof, it is, as mentioned before, parsimonious to do so until such time as it can be shown to be an incorrect assumption.
Those who make this assumption do so because it tallies with their experience of reality, and has yet to be shown to be false.
Can such a philosophical idea be proven?
I'm not sure.
3)
"Processes of geological change have always been operating in the past at the same rate, frequency and power as today."
I think this is open to considerable debate.
Not being a geologist but surely the solidifying of Earth's inner core has had an impact on the geology of our planet - whether frequency, amplitude etc?
But again, it is probably a reasonable parsimonious assumption to make, without proof, until such time as other data requires a revision in the assumption.
Has it been proven, or can it be proven?
To the first: not that I know of.
To the second: what would proof look like?
