What is the relationship between our conceptual understanding of mathematics and the physical observables it is so often used to describe? Let us take the hypothetical example of an isolated mind, devoid of any sensory input. Is it possible that such a mind could develop forms of mathematics? (A more involved question would be whether such a mind could have any thoughts at all, but this is a topic for another thread. In this case, we will assume that such a mind can have clear, succinct thoughts.) When we are initially taught the form and contexts of numbers, we are inevitably referred to physical objects that can have clear and independent conceptions in the mind i.e. 1 apple, 2 apples, 3 apples etc. But in counting the apples we are essentially describing a repetition of the ‘appearance’ of a clear and distinct idea in the mind or consciousness. Is it not possible then, that the isolated mind can have clear and distinct thoughts and experience a repetition of such thoughts, thereby raising the possibility of a conception of “counting�
From counting, many other kinds of mathematical operations can evolve, such as addition, multiplication, powers etc. It is also possible that division can be conceptualised (just being the opposite of multiplication) and therefore, it is also possible for non-integer numbers to abstracted. However, it is difficult to imagine irrational numbers such as pi and the exponential being conceptualised, seeing that they cannot be expressed as a simple division involving two numbers. Or is it? The irrational numbers such as pi and the exponential can be calculated from series extending to infinity solely involving integers, as can be seen in the websites below:
So then, if we extend this mental experiment a little further, let us give this isolated mind a mental capacity far exceeding our own and a very long time. From the simple conception of counting its own repeating thoughts, it is theoretically possible for this mind to generate all the necessary functions that describe how the observed extended world behaves, from wave equations to exponential growth rates to differential equations. So then, this mind that is isolated from the physical world can conceptually describe the mathematical functions that the observed physical world seems to yield to.
I don’t think a mind without any stimuli can do anything, because it must have something to react to, in order to think.
Perhaps it can be aware of itself? But even then, concioussness of one solitary thing is not enough to found mathematics or even necessitate math, I suppose. What good is saying there is only one of something if you cannot understand what having more than one of something, or even none of something, means?
“A more involved question would be whether such a mind could have any thoughts at all, but this is a topic for another thread. In this case, we will assume that such a mind can have clear, succinct thoughts.”
Perhaps I should have been more clear with what I was trying to demonstrate with the above. My post was not meant to question whether such a mind would find ‘meaning’ in developing mathematics, or what it is to understand having more than one of something, I was urguing the possibility of whether it is theoretically possible to build up the mathematics of the physical world, without knowledge of the physical world. I was using the isolated mind example to show that counting a repetitive thought could be enough to built up mathematics thereafter.
I find it to be an interesting conclusion, if valid
The basis of maths is something that’s been pre-occupying me for a few weeks now, so I was delighted when I saw this thread
IMO, the basis of mathematics is space. I think that once a set of axes has been set up, maths immediately follows. (By analogy, when the rules of chess were devised, the world of chess-possibilities immediately sprang up.) In cartesian space, lines and figures “naturally” appear, which give rise to the notion of adding and multiplying. For example, the law of commutation (AB = BA) is explicitly visible in the geometry, as are things like (a - b)^2 = a^2 + b^2 - 2ab. And calculus comes from space/geometry too - the fundamental definition is geometric. I’d wager that all other maths can be traced back to simple space. It follows from this that mathematical methods and laws are discovered, not invented.
In a nutshell: Space → Maths. And to be fancy: the Platonic Supersphere is in fact the a priori concept of space, as found in animals.
I do not believe that mathematics has to be solely based from space. I do believe, however, that it is easiest to derive mathematics by observing spatial relations. That is how we as a species have derived it, because we are very visual creatures. Logically however, there is no need for mathematics to be based on spatial relations, as I was trying to demonstrate above.
In my thought counting example above, there is no need for any conception of space, just an awareness of repitition. Once independent symbols of thought are given to different numbers of repetition we have counting. From counting, we can conceptualise addition, subtraction, division, multiplication, powers etc. In fact, I believe all of the primary mathematical operations can be derived just from counting. I have also shown above that even the irrational numbers such as e and pi can be derived from infinite series of integer operations.
So, if we gave this powerful isolated (and very curious) mind a very long time, by following all the possible logical connections of the manipulation of numbers, contained in these (practically infinite) logical connections would be all the mathematical laws that we currently use to describe the physical universe around us. Therefore, it is of my opinion that mathematical logic can be independent of extension.
First of all, I hope you won’t mind me not talking about this hypothetical mind, coz to be honest I can’t think of anything worth saying…
Back to the root of maths. I agree that spatial axes make “awareness of repetition” possible (and hence maths) but I’m not sure how these repetitions could start-out independantly of them. Maybe this is just me being blinkered by my inescapable concept of space.
The basis of all mathematics is logic. It has been postulated that a perfect grasp of formal logic would allow one to deduce mathematics. Unfortunately I’m deficient in formal logic, so I have to take my shoes and socks off to count past 10…
Yes, …and no. As long as we accept Newtonian physics and the ‘mechanical clock’ universe the answer is yes. As soon as we introduce quantum mechanics and the uncertainty principle, the answer is no.
Even in my dum-dum understanding I love quantum. It more nearly matches my understanding of a processual universe governed not by known (or yet to be known) principle, but with sponteniety and novelty as the coming into being and returning. Who knows the way of heaven?
Well I didn’t really want to get into our usual knowing versus not knowing debate, I believe each of us understands the other’s position and more, we accept the differences. My question was; "What if our very thoughts are made up of numbers themselves? What would that mean for thoughts. And what would then be the function of thoughts?
Well, if we look at numbers as representative of some kind of order to the universe, perhaps the grounding of the universe itself, then I suppose we can postulate a connection between our thoughts (using your assumption that they are also so ordered) with the very basis of reality…the stuff that makes up existence. That stuff might very well be our thoughts. Our thoughts are what constitute existence in other words.
No problems, I just introduced it to help clarify some of the ideas presented.
Spatial axes are conceptions of repeatability also, i.e. x number of units on the x-axis, y number of units on the y-axis. Therefore, in a conceptual sense, the awareness and abstraction of repeatability must be prior to an awareness of quantitative spatial axes. Now, one could argue however, that for there to be a conception of repeatability, there has to be a mind to conceive it. If we consider the mind to be simply an extension of the physical brain, then yes, this is true. However, the mathematics conceived in a mind does not necessarily have to involve a spatial awareness. The very basis of mathematics, in my opinion, is the simple conception of repeatability, or counting. Mathematics then flows in logical steps from this point on.
liquidangel,
Well, I think the answer to that question lies in your definition of what a number is, and how it comes about. Numbers are just labels that we have given to express quantities or repitition. In that sense, numbers are human abstractions. If you were to consider our thoughts to be made of numbers, then in my opinion that would require a higher level mind to have conceived of us, and given our thoughts numerical properties. In our level of awareness, our thoughts are not made of numbers, but in some higher, indepedent, level of awareness (if such a higher level exists) I suppose it is possible our thoughts may be conceived as such.
Ah ok. Yes my definition would not be the number itself but the quality represented by the number. I suppose I was asking if there is an order inherent in the substance of our thought. I believe there is but at the present moment I have no way of even beginning to understand thought at this level.
I’d rather differ with you on this one, Noely.
Say, you’ve got about 305 posts right now, after about 18 months on the forum. That number is indicative of a quantity and gives close no other information about you or your line of thoughts - it just expresses a mode of your appearance in time and it groups under a common symbol a mass of entities that belong to the same category.
It’s no secret that we associate numbers in reference to quantity in time or space and apply them to empirical data every time this becomes necessary. However, I don’t share the opinion that the sensory experience of repetition is what breeds the abstraction of numbers at the level of our intelect. If it were so, then pure arithmetical calculations involving only the concept of numbers would be stripped of their universal valability and could only be associated to objects that draw nigh to our immediate experience. You have to agree, then, that pure mathematical propositions are escorted by an intrinsical necessity - one could say they are a priori, as opposed to empirical.
Mathematical judgements, as I can understand, are synthetic. By saying that 1+2=3, one enlargens his intuitive concept of unity by adding a couple more, in order to grasp the concept of the newly formed number. That is, it is rational thought that commands numbers, and preferably not the other way around.
I don’t know about that. The line of our thoughts is largely inconsistent and usually lacks a definite direction. The father of modern Linguistics, F. de Saussure, affirmed in his Cours de Linguistique Generale, that “taken as it is, thinking is like a nebula where nothing is necessarily delineated.”
Form, in its strict sense, is necessary for an idea to take birth. To speak is to thimk, or, how Paul Valery put it, “a word doesn’t mean something, it means someone thinking about that something”. Considering the parameters of language, it is safe to say that being the body of an idea, the word could be transposed as a sequence of numbers, which is its inherent order. But, if crude thoughts are roughly a preamble to any expressed idea, lacking coherency, then we might have to wait more to gain an exhaustive knowledge of they link and jump off one another…
Numbers must be human abstractions. Do we see the number ‘46’ as an independent entity existing independently from a human mind? No. It was born and is understood (receives its ‘meaning’) in a human mind only. Numbers have been created by human minds to account for the repeatability of conscious ‘thoughts’ or intellectual entities. If we see two rocks on the ground, the mind will recognise a repetition of the conception of “rock”, and will assign the label 2 to the amount of times it has been repeated. We may argue that the rocks exist independently of the mind, but the conception of the rocks and the association of a number to the rocks exists solely in the human intellect.
Please re-read my initial posts. I have been arguing that the conception of numbers and repeatability (and thereafter mathematics) need not have anything to do with emperical or sensory input. All that is required is a conscious recognition of repetition (which in my isolated mind example, could be just a repetition of a conscious thought) which is not necessarily connected to anything physical. From my reading of the above paragraph, you are arguing the same (if not, very similar) thing as I am and saying you are disagreeing.
Mucius Scevola:
I can’t quite grasp the meaning of your last post (I’m sometimes a bit slow). I feel like a man who can see something glowing in a prickly bush, but who can’t quite navigate the prickles to get close enough to see it clearly. Could you explain it in the form of a reply to my first post (the 4th one) in this thread?
NoelyG:
I agree with you about on repeatability, but am still more-than-half of the view that in animals, spatial awareness provides the basis for this. Do you think that our capacity to do maths comes from something “deeper” and that our sense of space is subsequently “pencilled on” by these already-existant concepts?
I’ve been puzzling over this thread for a while and I must be missing something. While mathematics and/or logic can explain much of natural phenomena, are you suggesting that all that is can be reduced to numbers? Somewhere I got the notion that this was Russell’s failure in Principia Mathematica, even though he attempted any number of work arounds, he ultimately failed to show that logic could account for all natural conditions. I’m not aware of any breakthroughs in this issue. It seems like a bit of a stretch to say that the whole of human thought can be reduced to numbers.
