Is mathematics a mental idea?
Mathematics is an abstraction, an idea, a thought, a meaning and is not physical.
According to this answer, a mental idea cannot exist without a mind.
It depends on who you ask. Descartes would disagree with his dualism. Daniel Dennett would disagree with Descartes. Theology still tends towards dualism since it preserves supernaturalism, while cognitive science largely rejects it.
If mathematics is a mental idea, what does this imply about the laws of physics which can be modeled mathematically? Is there a way to fully explain this without resorting to a conscious mind behind them (either within us or more likely, outside us)?
It implies that consciousness generates meaningful expressions in mathematics and science and uses linguistic elements like syntax and orthography to describe, explain, and predict physical phenomena. Consciousness seems to be required for theory generation. Ever read a theory determined and expressed by an unconscious mind?
These are generally accepted ideas among contemporary philosophers of mind. Where they disagree is on meaning, terminology, and methods when confronting problems like the hard problem of consciousness.
This evokes metaphysical positions such as naive realism, transcendental idealism, and philosophical realism, etc.
If mathematics is a mental idea, what does this imply about the laws of physics which can be modeled mathematically? Is there a way to fully explain this without resorting to a conscious mind behind them (either within us or perhaps outside us)?
It depends on your metaphysical presupposition. If you believe in physicalism for instance, especially a non-reductive form, you according to Jaegwon Kim, you accept three principles at a minimum:
- The mental supervenes on the physical.
- The mind-body duality is a category mistake.
- The mental is wholly dependent and characterized by the physical.
(See Ch.1, Philosophy of the Mind)
As such, the "laws of physics" is both a symbol that given the triangle of reference refers to patterns in observation of phenomena that correspond with patterns in noumena, and das Ding an sich presuming a dichotomy between the observer and the external universe.