Meanwhile, what do you mean by "scientism"? I would say scientism is the belief that science is the only way to know whether something is true or real.
Scientism
...is the belief that science and the scientific method are the best or only way to render truth about the world and reality.
While the term was defined originally to mean "methods and attitudes typical of or attributed to natural scientists", some scholars, as well as political and religious leaders, have also adopted it as a pejorative term with the meaning "an exaggerated trust in the efficacy of the methods of natural science applied to all areas of investigation (as in philosophy, the social sciences, and the humanities)"
That puts you in the 'hard scientism' camp, which one poster here thinks isn't real and is only 'in your head' as opposed to 'real' science.
Now that we have this actual knowledge of what scientism is (as apposed to what some people
think it is) we begin to see the problem. By insisting that science (ie. the scientific method) is the
only way to know if some 'knowledge' is true or not, you imply that:-
1. Other methods of gaining knowledge are inadequate - which clearly isn't true. In many cases other methods are
better, because they produce useful results without so much time and effort. Often the science isn't 'settled' until the knowledge is already in regular use.
2. It is inherently far more trustworthy than other methods which cannot tell you 'for sure' whether some knowledge is real. But science is only as trustworthy as its application, which is not comprehensive or faultless. In some fields the science is so
unsure that scientists themselves have large disagreements, and what little 'knowledge' they do agree on is inadequate for practical application.
'Scientism' is often used pejoratively against those who elevate science to being the gatekeeper of all 'true' knowledge, dismissing everything else as untrustworthy or false. That is true in
some cases (eg. religion) but not all. The OP asks about logic, mathematics, and philosophy, all of which are
used by science but are not themselves science.
For example you can use logic alone to show that the supernatural doesn't exist,
without the need to gather evidence, develop theories, and test them. You may say that without that process I can't be
sure that eg. Jesus didn't perform miracles and come back to life as a god, but it works well enough in practice. Science OTOH has an impossible job trying to 'prove' a negative via lack of physical evidence and ability to test it.