Actually, it seems extremely unlikely that words would be required for symbolic thinking, considering that any animal advanced enough to base its actions on thought rather than pure reflex would need to have some kind of symbolic representation of the world.
Concepts exist without words, since words are just one part of a concept, and people with left temporal brain damage can lose access to a word without losing access to the concept.
A “symbol” sometimes means something atomic, which concepts are not. We probably have no symbols in our brains, in this strict sense.
Interesting point. I certainly agree that concepts/words are not actually atomic, or Platonic ideals, or anything like that. Concrete concepts, in particular, seem to correspond to “empirical clusters in thing-space”, or probability distributions over classes of objects in the real world (though of course, even objects in the real world aren’t really atomic).
Despite this, most people still view themselves as thinking symbolically, and many people believe themselves to be logical reasoning agents. After reading the first couple chapters of Jaynes, I am very convinced that the mind works probabilistically and does not actually deal with absolutes. Yet at the level of conscious reasoning, we seem to perceive the world in terms of symbolic absolutes. It seems like this could be either verbal or visual, but either way I have difficulty imagining conscious reasoning without symbols, even if more complicated clusters or probability distributions underly those symbols at a subconscious level. I wonder why this is.
Concepts exist without words, since words are just one part of a concept, and people with left temporal brain damage can lose access to a word without losing access to the concept.
We use words to solve word-problems. Before words, people thought without words but not about word-problems. Asking for techniques for solving word-problems without words is like asking how to create potteryware without pottery. There might be some sort of isomorphism that would allow you to think about words without words, but you would mainly end up with an encipherment. That is why I asked, “Has anyone developed techniques to do math without symbol manipulation?” We use words precisely because we need to know and use their systematic properties, in a similar way that we manipulate symbols to do math.
I am not saying that we are not able to understand the meaning and referents of words without words.
Geometrical constructions? If you have an accurate visual intuition of the properties of triangles and squares, then a diagram of the Pythagorean theorem pretty much is a proof of it. Euclidean geometry is an axiomatic system, but it isn’t a formal system on strings of symbols; it’s a formal system on abstract geometric figures.
Throughout most of history, math wasn’t done in what we now think of as “mathematical notation”, i.e. expressions written symbolically. That wasn’t invented until the 1500s. Before then, math was done with proofs written in ordinary language, accompanied by diagrams.
Has anyone developed techniques to do math without symbol manipulation?
Sometimes, when I’m doing certain kinds of calculations in my head, I imagine a clock and mentally manipulate the hands until I can see the “size” of the value I’m trying to reach (only then do I transform this value into a number). Is this similar to what you are talking about?
Yes. My OP asked about how to think without words. Your comment assumed that I had asked how to solve word problems without words. So I inferred that you thought thinking was a subset of word problems.
Has anyone developed techniques to do math without symbol manipulation?
This article presents evidence that symbols exist in our minds independent of words. http://artksthoughts.blogspot.com/2009/07/concepts-cognition-and-anthropomorphism.html
Actually, it seems extremely unlikely that words would be required for symbolic thinking, considering that any animal advanced enough to base its actions on thought rather than pure reflex would need to have some kind of symbolic representation of the world.
Concepts exist without words, since words are just one part of a concept, and people with left temporal brain damage can lose access to a word without losing access to the concept.
A “symbol” sometimes means something atomic, which concepts are not. We probably have no symbols in our brains, in this strict sense.
Interesting point. I certainly agree that concepts/words are not actually atomic, or Platonic ideals, or anything like that. Concrete concepts, in particular, seem to correspond to “empirical clusters in thing-space”, or probability distributions over classes of objects in the real world (though of course, even objects in the real world aren’t really atomic).
Despite this, most people still view themselves as thinking symbolically, and many people believe themselves to be logical reasoning agents. After reading the first couple chapters of Jaynes, I am very convinced that the mind works probabilistically and does not actually deal with absolutes. Yet at the level of conscious reasoning, we seem to perceive the world in terms of symbolic absolutes. It seems like this could be either verbal or visual, but either way I have difficulty imagining conscious reasoning without symbols, even if more complicated clusters or probability distributions underly those symbols at a subconscious level. I wonder why this is.
We use words to solve word-problems. Before words, people thought without words but not about word-problems. Asking for techniques for solving word-problems without words is like asking how to create potteryware without pottery. There might be some sort of isomorphism that would allow you to think about words without words, but you would mainly end up with an encipherment. That is why I asked, “Has anyone developed techniques to do math without symbol manipulation?” We use words precisely because we need to know and use their systematic properties, in a similar way that we manipulate symbols to do math.
I am not saying that we are not able to understand the meaning and referents of words without words.
Geometrical constructions? If you have an accurate visual intuition of the properties of triangles and squares, then a diagram of the Pythagorean theorem pretty much is a proof of it. Euclidean geometry is an axiomatic system, but it isn’t a formal system on strings of symbols; it’s a formal system on abstract geometric figures.
Throughout most of history, math wasn’t done in what we now think of as “mathematical notation”, i.e. expressions written symbolically. That wasn’t invented until the 1500s. Before then, math was done with proofs written in ordinary language, accompanied by diagrams.
Sometimes, when I’m doing certain kinds of calculations in my head, I imagine a clock and mentally manipulate the hands until I can see the “size” of the value I’m trying to reach (only then do I transform this value into a number). Is this similar to what you are talking about?
What would math without symbols consist of?
What would language without words consist of?
Do you believe that people who haven’t been taught language, can’t think?
You are putting words in my mouth. Have you read my other comment?
Yes. My OP asked about how to think without words. Your comment assumed that I had asked how to solve word problems without words. So I inferred that you thought thinking was a subset of word problems.