All software can be reduced to mathematical formulae, which cannot be patented, so no. There should be no such thing as a software patent.
However, any physical machine or process can be reduced to the laws of physics, and you can't patent the laws of physics as far as I know. I don't think "reduced to mathematical formulae" is in itself a great argument.
Should the jacquard loom have been granted a patent?
It seems like the old pattern of "Patent a certain method of weaving brocade", then later "Patent a certain method of weaving brocade...on (mechanical) computers!"
To the original objection that you can't patent math, I think some countries do disallow software patents, don't know if it's that reason or something else. But if they can disallow because math, shouldn't they disallow all patents because physics?
German patent law for example has had the term "Technizität" as a neccessary property for a patent to be valid.
https://de.wikipedia.org/wiki/Technizit%C3%A4t
Money quote from the BGH (German Supreme Court): "Als patentierbar anzusehen ist eine Lehre zum planmäßigen Handeln unter Einsatz beherrschbarer Naturkräfte zur Erreichung eines kausal übersehbaren Erfolges." (GTrans:"To be considered patentable is a doctrine of planned action using controllable natural forces to achieve a causally overlookable success." :) )
i.e. You gotta use physics in your patent.
I have been coming back to this time and again, and today I still see it as the least shitty definition in regard to what should be patentable and what shouldn't.
Other national patent offices in Europe have used similar demarcation properties and it had served them quite well...until the degradation of patent standards started here as well around the turn of the millenium, with the first act being the EPO starting to issue software patents in rather blatant violation of its mandate.
Patents exist to promote novel applications of physics and chemistry, but do not forbid use of the fundamental laws, and should not forbid use of fundamental mathematical or algorithmic processes.
If you can violate a patent in your mind, it shouldn't be granted.
In Europe at least maths can't be patented but applications of maths often can, just like software can't be patented as such but if it has real-world technical effect it can.
A new rocket launch trajectory may be "just maths" but if it reduces fuel consumption then it arguably isn't "just maths" and so will probably gain a patent.
This approach seems compatible with the quid pro quo intention of patents.
Only because we've decided so. In the United States, for example, you can copyright the law.