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Is mathematics independent of human consciousness?
in Philosophy
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This is a tricky one, and the answer strongly depends on the assumptions made. For example, we can assume that the "objective reality" simply does not exist, and only the reality skewed by our perceptions does - and then, of course, mathematics will be inseparable from human consciousness.
I like to propose the following mental experiment to make such discussions more concrete. Suppose the first contact with an alien species occurs some time into the future, and suppose it is peaceful, that is, instead of starting a war with each other, we communicate and learn from each other - a mutually beneficial partnership.
One of the first things we will want to discuss, of course, is science: we will want to improve our technology by utilising the knowledge the aliens have and we do not. And inevitably we will get to talking about the "sciences above sciences", such as mathematics, on which many other sciences are based.
Will our mathematics be similar? Of course we will use different notations, formulate different theorems, etc. But at the core, will our mathematics be "translateable" into each other, or will they be principally different and incomprehensible to any species other than us?
There is a very complicated discussion to be had here. I do believe that all intelligent organic species' development must be somewhat similar, since fundamentally they all are constrained by shared goals such as acquisition and utilisation of resources - but it is possible that sciences in different species are developed in different ways, especially if the order of discoveries is different from ours and, say, they manage to develop a gas before a steam engine. It is very much possible that, in order to describe their discoveries with a metalanguage such as mathematics, they will have to start from completely different axioms, leading to the mathematics that describes the world in a completely different way.
Theoretically, there is infinity of ways to interpret the observations of the world around us. It is very much possible that two species will have completely different concepts in mind; say, we see the world as fundamentally consisting of elementary particles, while the other species might not acknowledge the existence of particles and, instead, see everything as uniform matter distributed in a peculiar way. Mathematics can differ just as much; where we see parallel lines, someone else might not acknowledge their existence as in the real world "infinite lines" only exist as an abstraction and not real physical objects.
To summarise, I think that mathematics is partially (and, likely, mostly) dependent on the species' consciousness, but not entirely.
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Do we humans invent mathematical forms as we need them and then merely discover their emergent properties later? Or are those mathematical forms innate to nature, and are hence discovered rather than invented?
The most logical way we could communicate with another civilization would most likely be through maths
If there's even one technological civilization elsewhere in our unimaginably vast universe, then that civilization must have discovered enough math to produce technology.
If math depends on human consciousness, then we're the only technological civilization in the universe which I don’t believe to be the case
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The way we build mathematics is by stating a set axioms, and then discovering where those axioms led us. So I would characterise the development of a given mathematical field as a process of discovery, rather than invention.
However, what axioms we assume to hold is somewhat arbitrary. If we choose a different set of axioms, then the process of discovery will lead us to different conclusions, often to the ones that contradict those resulting from the original set of axioms.
It is possible that other civilisations have different axiomatic approaches to mathematics, leading to conclusions which to us, humans, look bizarre and alien (pun intended). It even possible that other civilisations interpret logic differently, so they build different causative connections. And moreover, it is possible that other civilisations do not use the same terms as we do - what if the concept of "number" is meaningless to some species, that use a completely different concept when performing counting?
I do think that all valid sciences developed by all species must be inherently translateable into each other, since they describe the properties of the same Universe - but such translation may be incredibly difficult, if not impossible, in practice, when two species with principally different evolution histories are involved.
As such, I would answer your question as follows: "Living beings invent mathematical forms and adapt them to the innate mathematical forms of nature, and the details of that adaptation depend on the beings". So, in a way, mathematics is both innate to nature, and developed by us humans.
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>However, what axioms we assume to hold is somewhat arbitrary.
I don’t accept that an axiom or postulate is a statement that is taken to be true, to serve as a premise or starting point for further reasoning and argument
> If we choose a different set of axioms, then the process of discovery will lead us to different conclusions, often to the ones that contradict those resulting from the original set of axioms.
You will have to give examples of this please
>It is possible that other civilisations have different axiomatic approaches to mathematics, leading to conclusions which to us, humans, look bizarre and alien (pun intended). It even possible that other civilisations interpret logic differently, so they build different causative connections. And moreover, it is possible that other civilisations do not use the same terms as we do - what if the concept of "number" is meaningless to some species, that use a completely different concept when performing counting?
Math to me would hold universal truths the only difference being the language, 1+1 will still be 2 and a circle always a circle
>I do think that all valid sciences developed by all species must be inherently translateable into each other, since they describe the properties of the same Universe - but such translation may be incredibly difficult, if not impossible, in practice, when two species with principally different evolution histories are involved.
Lost tribes round the world still had a basic form of math that was comprehensible to us
As such, I would answer your question as follows: "Living beings invent mathematical forms and adapt them to the innate mathematical forms of nature, and the details of that adaptation depend on the beings". So, in a way, mathematics is both innate to nature, and developed by us humans.
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Is mathematics independent of human consciousness?
I agree with @MayCaesar that Maths are not a creation but an ongoing discovery... I would include Logic in the greater notion of Mathematics (or vice-versa) as they are quite intimate little concepts IMO...At a very fundamental level, I think that it is ingrained in the very fabric of this deterministic universe, it expresses itself in recognizable patterns all across the universe, from the smallest molecule to galaxy clusters configurations. Come to think of it, it's actually the very platform on which rests Space-Time... Without Maths, there's no universe...
@Dee
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