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Say you were to come upon a mathematical equation whose variables and sheer complexity were beyond your knowledge at that point in time. As an example, a simple algebraic problem may seem to be a completely different language to young children who have hardly mastered their multiplication tables. Note that I said an "equation" and not a mathematical problem.

This means that you see two sides, neither of which are wholly understood by you, that have been determined to be equated.

Despite the fact that you don't understand it, you continue to build your knowledge based only on what you have already learned, and one day, the equation finally makes sense.

In a realist or solipsistic view, reality is either generated by your own mind or illusory altogether. How, then, can you 'think up' an equation or be presented one illusorily, that makes no sense to one's self, yet build on rules that you had previously known (or, rather, believed to have known) to determine that the original equation is true?

  • Hi; well, it might be one good reason to see that Solipsism isn't generally taken to be a rational (ie good) account of reality. – Mozibur Ullah Sep 4 '15 at 1:15
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    in the last paragraph, you seem to be conflating realism and solipsism in the last paragraph. – virmaior Sep 4 '15 at 1:35
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    I don't see how you can think up something in math which does not make sense to you. You may not be able to analyze everything about it right away, but it would still "make sense." – Daniel Goldman Sep 4 '15 at 2:01
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    I'm not at all following how you're using the word "realist" then. – virmaior Sep 4 '15 at 8:32
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    Do you mean idealism rather than realism? – virmaior Sep 4 '15 at 14:23
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See Jabberwocky; it is a "well written" nonsense poem, full of suggestive rhymes and words, like :

All mimsy were the borogoves.

It is readable and enjoyable, and it has beeen translated multiple times.

So what ? From a solipsistic point of view, how it is possible that your own mind can 'write up' a poem that makes no sense, yet built on syntactical and rethorical rules that you are mastering ?

Thus, if you do not "understand" it, someone outside you must have "created" it, and thus solipsism has been defeated...

This looks like a variation of Descartes' argument base on doubt.

  • This is, perhaps, a better example than my own. It's funny; I recently re-read Descartes' meditations and sort of came upon this based on Descartes' doubts in his first and second meditations. – Goodies Sep 5 '15 at 3:13
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Given Allegranzas excellent answer, this is solely a clarification on Virmaiors comment on confusing realism and solipsism in the following statement:

In a realist or solipsistic view, reality is either generated by your own mind or illusory altogether.

This is somewhat confused; there are two basic positions on reality - realism or idealism.

Realism, takes the world to be an objective fact and has nothing to do with the mind; this does not mean that there are no minds; but that they have nothing to do with the construction of reality, or in understanding it: positions that go along with this view are various forms of materialism - one flavour being Physicalism.

Idealism, takes that the mind is in some indispensable way involved in how reality is to be understood; for example Kantian Transcendental Idealism; or the 'brains-in-vats' scenario, philosophically has nothing to do with science-fiction, but with the position all there is are minds, and minds not in the singular but in the plural; and solipsism, which takes the opposite view, that there is only a mind in the singular; but more, not just any mind but specifically my* mind.

Thus your statement should read: in solipsism, reality is constructed by your mind.

It's worth noting that when reality is taken to be illusionary, this is because it's suggested that there is a deeper structure or unity to reality that is being missed; for example compare the veil of Maya in (Vedantic) Indian philosophy.

  • Wow. My apologies. I meant "idealism" almost this entire time. What a boner on my part. – Goodies Sep 5 '15 at 3:25
  • @goodies: you're welcome; it's perhaps worth editing your question to take that into account ... – Mozibur Ullah Sep 5 '15 at 3:59
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I like Mauro's analogy, and Mozibur makes a good point about the nature of realism.

I'm not sure this is entirely what you have in mind, but it may be of interest nonetheless.

You ask : "How, then, can you 'think up' an equation or be presented one illusorily, that makes no sense to one's self, yet build on rules that you had previously known (or, rather, believed to have known) to determine that the original equation is true?"

This is not uncommon in both mathematics and physical science.

For example, the method of analytic continuation used to extend the domain of analytic functions in the complex plane is a well-defined, rigorous, rule-based method of complex analysis. It is a method that is well understood by mathematicians, though not necessarily by me.

When this method is applied to the Reimann zeta function it yields :

ζ(-1) = -1/12.

In other words :

-1/12 = 1 + 2 + 3 + 4 + ...

How is one to make sense of this equation. How can the sum of all natural numbers equal negative one over twelve?

The odd thing about this identity, as I understand it, is that it occurs in our physical theories. For example, I believe that it occurs naturally in string theory. There are also other ways of deriving this identity. The notable Indian mathematician, Srinivasa Ramanujan, derived this identity in the early 20th century. Other methods also produce the same result.

Perhaps we would do well to make note of Hegel's position that, when we apply legitimate methods to obtain seemingly absurd results, we must accept the legitimacy of those results.

EDIT

Here are some examples of the identities Ramanujan sent to Hardy from India before Hardy invited him to Cambridge. These identities were derived without any knowledge of the methods of complex analysis.

Ramanujan's identities

  • I fear I may not have been as clear as I had though. I am aware of the rather odd implications of mathematical infinities such as the Reimann Zeta function of -1. However, this is not what I have in mind. A hard psolipsist would claim that he is, indeed, the only mind in existence. An idealist would state that reality is generated by the mind. That being said, if you were to see an equation that did not make sense, it follows that the equation originated in your own mind. Not the interpretation or sight of the equation, but the thing to which the equation refers. – Goodies Sep 5 '15 at 3:10
  • (cont) With that, how can that equation, which was, for all intents and purposes, invented by your own mind, turn out to be true? Despite the fact that you did not consciously understand it, it's as if your mind was able to 'jump forward' and generate this equation prior to having the knowledge to do so. – Goodies Sep 5 '15 at 3:11
  • @Goodies Perhaps this was the case with Ramanujan. His mind created these identities with only the most elementary of mathematical teachings, and yet his mind was able to make the leap forward to state such identities. He was apparently unable to explain their origin to others or to justify their truth, and it is not clear what he understood their meaning to be at the time he produced them. Hardy initially dismissed them as school-boy errors. – Nick R Sep 5 '15 at 3:18
  • That's not exactly what I mean. Say a small child sees the Schrodinger equation for the first time. With absolutely no concept of even rudimentary quantum mechanics, their mind was still able to generate that formula within their perceived reality. The forumula itself, written on paper with graphite, represents something much greater. That child later grows up to be a physicist and is easily able to understand the equation and its implications. Did they merely generate reality to fill that gap to turn something that was actually purely randomly generated into something that makes sense? – Goodies Sep 5 '15 at 3:24
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    @Goodies Oh, I see, yes. The random element would certainly be missing in the case I have cited. – Nick R Sep 5 '15 at 3:27
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Knowledge is two-fold; there is knowledge from observation – things we see, the equation, and there is knowledge as in understanding, which is derived from observation. It is a loop, going from observation to understanding and back to observation.

The equation is the same, but at some point you'll have ammased enough derived knowledge – or understanding – that you will see the equation and understand it.

  • How would one attribute mathematical knowledge to observation? – Nick R Sep 6 '15 at 22:17
  • What is mathematical knowledge? If you mean individual mathematical knowledge, then we learn through observation. – Wilhelm Friedrich Sep 7 '15 at 20:41
  • How does one observe the set of natural numbers, or the knowledge expressed by the theorems of Category Theory? – Nick R Sep 7 '15 at 20:57
  • I think this takes us back to my initial statement, where i describe the loop that allows us to build knowledge. I would classify category theory and natural numbers as models we have build and observed to adhere to the truth. To add perspective to my original statement I'd like to draw on the notions of inductive and deductive knowledge – Wilhelm Friedrich Sep 7 '15 at 22:10
  • Fair enough. I guess that's one way of thinking about a very difficult subject. For me, I feel I am able to form knowledge as mental abstractions independent of observation. – Nick R Sep 7 '15 at 22:59

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