I am currently attending university and taking classes in a language other than my native tongue. There is always this continual inner push to spend more time thinking in the foreign tongue in order to increase my fluency, which vastly increased before, when I had had a very strong commitment to think in the foreign tongue, and whenever I noticed that I was thinking in English, I would spend the time to think the same phrase out in Spanish .
One day during a lecture I was tempting to understand a certain chemical reaction with multiple steps. When I had been at it for quite a while, I asked myself which language I was using? Then I realized I hadn't been using any language, but only the symbolic language of chemistry: arrows, chemical symbols, numbers that represent multiples of various chemicals. I thought this was odd for me, because I am someone who has much more talent in the area of language than symbolic reasoning (at least when applied to mathematics).
I was once looking at an animation of a pumping heart. I didn't know any of the terminology of the various parts, but had an understanding of the path that the blood takes, from the body into the heart, onto the lungs, then back into the heart and out to the body. I found the animation terribly fascinating, as I sought to follow the path of the blood without any words. It seemed to me like I was trying to understand the system not sequentially as a chain, but more as a whole. I was seeking to relate the state of each part of the system to the concurrent state of any and all other components at every given instance along the way. Not being familiar with the Spanish terminology, I found myself in a limbo trying to relate everything without being able to label the parts. I think it provides a valuable insight in that keeps you away from the sequential orientation of thought that language often requires, when you go on thinking without it.
I think this is something like what it takes to understand how the solar system really works. you have to see multiple moving parts in your mind all at once and understand how they all interact to foresee the resultant phenomenon like eclipses and the like. I think I am weak at this when it involves more than just two interacting factors. I have noticed other people have a lot more facility with this sort of thing, but it seems to me like something you can grow in with practice. I once heard that Einstein said he would experience thoughts in the form of some kind of inner animation and then put that into words and finally mathematics. I can't put down a source for that. Temple Grandin, a gifted inventor with autism, has said the same thing. Although I am not the best at mathematics, when working with chemistry I find a certain degree of facility based on being able to imagine molecules and the way they interact
Today I was walking and I found myself in a language limbo moment again. I decided I would try to extend it as long as I could, and even if I began thinking in words, I would just let it go and continue observing things around me. At the moment I saw some grass and thought green I knew I had faltered, but realized as the cars passed me by I had just been observing them interacting and the sun coming from the sky without making any attempt to label anything. It was very interesting, and it seemed to me to be something like holding your breath to maintain the state. At every moment that you realize you are in the state you want to make a statement in words about it and it takes a strange active passivity to then enter back into it again. If that makes any sense! Active Passivity, ha! Well, maybe I never did ever think without words and I was just fooling myself, but I think it did happen. The state seems to lend itself to a certain kind of unconcerned selflessness, but not in a traditional sense. It's more like, you can't see yourself thinking negatively or positively about anyone including yourself while you are in it. You can make no plans for good or evil. At least to me that was how it seemed. I wondered if it was somehow inconsiderate, or wasteful to inhabit that kind of thinking, like a misuse of the human mind. Finally I began to desire to pray to God. As I believe in God and am a Christian of the sort that believes the God described in the bible is the true one with both his corresponding actions towards man of Mercy and Judgement. I realized I couldn't connect to God without words. If something came into my mind about the nature of God how could I justifiably, in my attempt to go on without words in my mind eschew it. Finally, I tried to just present certain non verbal sentiments to my Lord, but it seemed inadequate and I cried out to him, Daddy. The next thought that occurred to me is that humans need language in order to communicate with God and that was maybe the main original intended purpose of it! Then the thought entered, "But maybe the word attitude signifies certain morally significant communications towards God that can be expressed on through to the heavens." Perhaps, but I think it is necessary to verbalize what you are able to verbalize.
Wordless thinking seems valuable, especially for mechanical reasoning, but to the extent that it isolates you, I think you need to abandon it, and also use language to communicate your non-verbally reasoned intuits to others including God himself. But right now it is occurring to me, I never really attempted to maintain that state of non verbal thinking in a forced or immersed social setting. Sex must be an incredible form of nonverbal communication. As well as all displays of affection or body language.
Another example of nonverbal thought that almost everyone has experienced, and probably more than ever a s small child, is the state in which you know what you want to communicate but have yet, or are unable to formulate the set of words which effectively communicates it. This often happens to people who are writing an essay and encountering complex ideas. Obviously you have formulated a thought, yet language is currently in a state of failure of expressing. So, for me, this proves that the state exists. Thoughts exist before words and are then put into words as a conduit to express them to others. Then, in the process of conforming thoughts to words we notice that we ourselves are provided greater insights into the phenomenon we are expressing! So, we then begin to speak to ourselves about everything whether doing so is beneficial, harmful, or indifferent. I think it could be any of the three at any given time! I noticed that after I learned the terminology for the parts of the heart it was much more easy to understand and put the process together as a whole, but I lost interest in that super understanding which would have been what I described earlier as seeking to relate the state of each part of the system to the concurrent state of any and all other components at every given instance along the way. Language seems to facilitate picking a certain point in time and answering questions about what is happening somewhere else in the system, as in taking a snapshot of the phenomenon. Thinking without words seems to work more on the basis of understanding multiple occurrences within a system and being able to evaluate the whole and each isolated part in a fluid, kinetic, more global way with less differentiation or isolation of the component parts.
Something which might be easier than completely eliminating the internal monologue would be to optimize it instead by adopting or creating a language with fast pronunciation, especially for the most common words. I looked through Wikipedia's list of conlangs but didn't find anything similar late last year.
I think this is an interesting question. What is it that you want to be able to think of non-verbally? Some of the replies discuss using this to perform mathematical manipulations. Is this your primary goal, or is it something else? What is it you want to process faster? Also, does this mean you want to be able to visualize more, or that you want to somehow think of things in an abstract but still completely non-verbal way (abstract meaning nothing that you could see, hear, touch, taste, or smell)? I may have some input for the first, but not really for the second (at least, not that I know of--the only things I can't see, hear, smell, taste or touch that clearly exist in my thoughts are my emotions, though these are often also imbued with linguistic and visual associations).
Being able to visualize doesn't necessarily require cutting off your word-thinking abilities at all. I know this because I think of a lot of things both visually and in words and symbols and tend to swap between the two interchangeably. This is actually my natural mode of thought. Whether you can adopt my approach may depend on how your mental architecture works. On the other hand, maybe it is possible to learn how to adopt this approach even if your mind doesn't currently work this way, much like people can train to improve their memorization skills. I don't currently know how easy or hard that is, but I like the approach of trying to change something to see if it works. As I understand it, some people are very good at visualization and some never think in pictures at all and are even surprised to learn that some people do. Some people can play songs in their head, and some people can't (I can do this, though I have no special musical ability). I know that I often use visualization both when I am doing math and writing novels, and also when I'm just daydreaming. On the other hand, I will sometimes both see and hear words and symbols in my head in addition to or instead of visuals.
What is the current way that your mind works? Are all of your thoughts in your personal voice, like hearing yourself speak inside your head, or are some annotated with pictures, sounds, smells, etc? People have very different ways of naturally thinking about things, so I'd need to know more to write a more detailed response to your question. I would also be immensely interested to know what kind of thinking styles people have here.
Everything. The words of my internal monologue play out slowly, all of them after the thought has formed. When I hear the first word in my mind, I already know the mental content of the sentence, though sometimes I get stuck along the way trying to pick a word out. Even then, I clearly already am accessing the concept for the word I can't find. A sentence may take 10 seconds to listen to in my head, but its complete meaning, and some general syntactic structure, seems to take less then one second to form. The words, as far as I can tell, serve no purpose when I'm not speaking to someone else. Yet I habitually wait for them to roll out before moving on to the next thought.
Being able to visualize things would be nice, but I have almost no ability to visualize things. I can't imagine my mother's face, or the front of my house; I can only recognize it. I have something like or analogous to visualization for vector spaces. I can often feel out how things move in a low-dimensional phase space via pattern-recognition rather than math, probably because I've spent so much time observing data which describes such paths. I have a tactile sense for type matches and mismatches; type mismatches (category errors) in spoken language stick out to me almost like a red dot on a blue field. I think my understanding of logical arguments and algorithms is also pre-verbal; I seem to grasp the logical structure of, say, code I'm writing or reading, before I can put it into words. I suppose this comes from spending tens of thousands of hours writing and debugging code. I don't know if any of these things are unusual. People don't seem to talk about them, though; and many people act as if they had no such senses.
I have no special musical ability
Oh?
One classic evil genie move is when someone wishes to be the best at something, it makes everyone but the wisher incompetent at it.
That illustrates a sense in which unique special isn't as special as we feel it is, unlike awesome special.
Hmmm. Okay. I'm not sure what you're trying to say, though. Are you trying to say that you think this ability is unique but not interesting or that it is interesting but not unique (I would say it is nice to have but not unique)? Is there a reason you have a particular interest in this ability, one way or another, or are you making a comment on the phrasing of the sentence? I looked at the articles you linked to, but the meaning of your comment is still unclear to me. Perhaps I am missing something obvious.
I am trying to say that the ability to play music in one's head without the presence of a musician or radio, etc. is a superpower, a form of mundane magic.
Granted that it is an awesome ability, how can we explain referring to this super power as "no special ability"? By noting common biases! People might think less of their musical superpower if they compare it to greater superpowers, such as the ability to freeze time, or the ability to compose music. Once one asks "how good is this superpower?", one might forget it's a superpower if the answer is "not comparatively good".
I see now. Thanks for the clarification (and upvoted because of it). The phrasing I used does seem to imply that I don't think that it is a special talent, although that was not my intent. I actually meant that I can play music in my mind but not on an instrument, nor can I write music (I can compose short melodies in my head, however). I do greatly enjoy being able to listen to songs I know by playing them in my head. Playing them on an mp3 player is a bit higher fidelity, but requires charged batteries, so my ability does come in handy.
Probably a more accurate thing to say would have been that I can hear or compose music in my head, but I do not have the ability to play an instrument, sing, or write music (at least not currently well enough that I would be able to pass any kind of music test whatsoever). I tend to be fairly happy with what I have, actually. While I do compare what I have to what other people have, I am rarely dismayed by the comparison. One reason for this is that other people often have rather different utility functions than I do, so I tend to not be trying to maximize what they are.
I agree that it is highly useful to appreciate what you have. Especially since that will help you enjoy life and solve problems better than the things you don't have.
I can't bring up an example, but I suspect that I've often noticed an error in a thought only after it was verbalized.
I did this a few years ago, but I'm not sure exactly how. I wanted to think less verbally because I worried that my thoughts were too constrained by words, which kept me at the very surface level of my consciousness and perhaps inhibited my access to deeper parts of my mind. I think that part of the transformation came about simply because I wanted it to (power of suggestion). It probably also helped that I started watching a lot more films and doing more math. I don't remember the exact process by which I transformed my thought-structure.
Something that I noticed was that my thoughts got much less verbal when I became more emotional. During a few particularly emotionally intense experiences this past year, I found myself less able to reason through thoughts verbally. At the time, I had an impression that my subconscious mind had taken over my thought process, closing my conscious mind out, and denying it the power of words that would let it interfere with the transformations going on inside me.
A few months ago, I decided to start thinking more verbally again, so that it would be easier to fulfill my dreams of being a novelist. I'll try to remember exactly how I did this, but the process is not wholly clear to me. I know that at some point I made a decision to "reprogram myself" to be more verbal, and I think that my desire to transform contributed significantly to the actual transformation. I also made a point of trying to express myself more verbally in my mind. One exercise I did involved looking at things in the world and trying to come up with eloquent verbal descriptions of them. I also started writing more in my journal, and reading a lot more. This wasn't a rigorous scientific experiment, and I didn't keep very careful track of the different things I was doing to reshape my thought-structure, but whatever I did worked, because I think very verbally now.
I'm not sure if this was helpful at all, but I figured I should comment here, since this is something I've actually done. I also don't know how similar these processes are between different people, or whether it matters that I'm female. Furthermore, I'll note that it was much easier to train myself to think verbally again than it was to make myself think less verbally; I was always a very verbal person growing up. Not sure if that's nature or nurture or both.
Has anyone developed techniques for thinking without words?
Michael Vassar and some associate of his did researchers into this. He reported some success but for him the without-thoughts thinking was just slower than the verbal form that he had been using for so long so it was not worth switching.
It isn't clear how much ability people have to think without language. Some people report having running monologues pretty much constantly, but others do not.
There are also a handful of famous cases of people who grew up with little or no language exposure. The most widely studied example is probably Genie. Another well known example is Ildefonso, described in Susan Schaller's book "The Man Without Words". Ildefonso was a deaf child who grew up in an environment where his parents had no way of communicating with him. He wasn't exposed to sign language until he was already an adult. That example makes one understand how people in ancient times considered deaf-mutes to be effectively stupid and crazy, although it isn't completely clear if that's due more to the lack of communication or the lack of language. In the case of Ildefonso, the story goes that when he realized that objects had names in sign language he burst into tears at the revelation.
Language is a very basic part of human thought processes. In order to do almost anything of note we need to be able to use language. I'm not at all convinced that developing ways of "thinking without words" is at all productive.
I'm not at all convinced that developing ways of "thinking without words" is at all productive.
I have the opposite testimony. The brain is very visual. Audio processing has pretty narrow bandwidth by comparison. It pays to use your brain's GPU.
By your logic deaf people taught sign language should be geniuses. Language is entirely separate from audio processing, that's just how we communicate.
Language is useful. (Though it would be less so if people didn't automatically assume being nonverbal made you completely stupid. Picture boards!) It doesn't follow that we have to think exclusively in words. Language is not a very basic part of Amanda Baggs's thought processes, for example.
Thinking without words is useful because it's (sometimes) fast, and has different constraints.
I've found that when I try and do basic arithmetic in my head (like 1784 * 2), I end up knowing the answer (or some subanswer of the algorithm) in some subsymbolic way but then feel the need to carry out the calculation with the symbols. I can calculate quicker by jumping straight to the answer I already know, although it feels like making a leap of faith. It sounds like you want to do a similar thing with words.
I end up knowing the answer (or some subanswer of the algorithm) in some subsymbolic way
Do you mean you have a subsymbolic concept of 3,568 or that you seem to arrive at it in a subsymbolic way?
I'll normally double things starting at the right and going left. So first 4*2=8, no problem. Then I need 8*2. As soon as I think the number 8 my brain instantly returns the number 16 as its double, but I still feel like I need to consciously think to myself "What's eight times two? Sixteen". Then the same thing happens for 7*2=14. Now I feel like I need to add up the work I've done so far. At this point my brain instantly returns the number 1568 (The number 1568 appears in my mind as a sequence of digits, not visualised visually, but just the concepts "one" "five" "six" "eight" in that order.) However I still feel the need to think the thought "16 represents 160. 160 + 8 = 168, 14 represents 1400, and 1400 +168 =1568" even though I already know that the answer will be 1568. Then 1*2+1=3 so we have 3568.
I don't think I visualise the numbers at every stage as their visual symbols, I just have concepts for each of the digits which I shuffle around. If I need to store a number to remember it later then I sub-vocalise it and then later I can just recall what I "heard".
There's this known result that people make decisions intuitively and later come up with verbal rationalizations for them but when all of that gets assembled into a stream of consciousness, it can feel like the rationalizations came first and were the deciding factor. Your experiecne could be something like this, only in the other direction. Or you have a really high-definition number sense. Do you still get correct results if you try to surpress the verbalizations while doing mental arithmetic?
If I don't verbalise the multiplications as I carry them out then I often forget or misremember the running total when I want to add another number to it. I feel like practice would improve this.
Wow, I'm glad I asked, since that is interesting. I think I can see how the computation could be automatized and regulated to subconscious components of your brain through practice, but its neat that your brain can keep track of those products (8 .. 16 .. 14 .. 2) and their place value (8 .. 160 .. 1400 .. 2000) and add them.
This kind of question is now open to experimental investigation, with the use of TMS (transcranial magnetic stimulation) to selectively and temporarily interfere with specific brain areas, for instance the language modules. Here is one hit from a Google search.
With a little more time and effort than I have available right now it should be possible to turn up some interesting facts.
Has anyone developed techniques for thinking without words? That is the question. Whether 'tis nobler in the mind to think in words. The slings and arrows of bewildering neural processes. Or to take arms against a manner of cerebration.
Enough of my soliloquy.
A good follow-up question is "How do people born blind think?" OR "How do people born deaf think?".
There is an answer out there.
Buddhist practices of emptying your mind
In my many hours of reading texts on Theravada Buddhism I do not recall hearing about such a practice. (7th or 8th jhanas, maybe?) I have read less about Zen but do not recall seeing anything very close to "emptying your mind" there either. I think Mahayana is the work of Satan and thus I have not studied its unholy scripture, but Wikipedia articles did not mention mind-emptying. Google searches of similar phrases yield pure garbage. No anchor can make it make sense for samadhi, and vipassana is right out, so... Are you referring to something other than the vague Western meme of 'emptying ones mind'?
Anyway, I was under the impression that math, programming, etc---abstract reasoning---were conceptual or symbolic but very wordless, and that generally intuition is wordless, but perhaps intuition is too fast by your criteria?
In my many hours of reading texts on Theravada Buddhism
Slight derail, but if one was curious to study similar things, what would you suggest they look into/look for? What would you recommend for someone who wanted to study Buddhism and avoid many of the western memes?
This is a useful resource. The Tipitaka is like a bajillion pages long and I'm not even sure that it's been wholly translated into English. If you're interested in the practice of Buddhism there are much better sources, but if you're interested in the spiritual philosophy of Buddhism then as far as I know diving into Wikipedia is probably the best place. My advice is to be wary of Mahayana or Tibetan memes and also to stay wary of interpretations of the original text that appear to be patently wrong. Also it is important to remember that oftentimes interpretations do not take care to keep the map-territory distinction in mind, which can result in misconstrual of the original meaning. Also it is important to remember that Buddhist and Taoist ontology is not Western. When the Buddha talks of "rebirth" he almost certainly does not mean the patently absurd thing that most Westerners imagine. Reading lots of Wikipedia articles should give you hints about what further texts you should read. accesstoinsight.org is pretty cool. This, for instance. Also, I find this to be concise, chilling, and a hint at the alien-ness of the Buddha's original message. I have a hard time reconciling Buddhism (or Taoism) with my metaphysical default of bland computational Neoplatonism.
Excellent testbed!
I had a hypothesis about my use of "internal speech" and used that game to give myself a nominal task to see if I heard any internal speech while engaging with it (and if so with what content). Prior to playing with the game, I had a sense that I rarely use elaborate and intelligible "internal speech" except in the context of high impact communication, as when rehearsing a speech for an audience, planning to write prose, retrospectively mulling over an emotionally fraught conversation, or trying to figure out what I'm capable of saying in a second language. In those cases I imagine saying/writing something to see how it sounds, and then imagine alternative things I could say instead.
For normal daily life, as near as I can tell, I simply think in images, kinesthetic feelings, and gestalt model/analogy/diagram/datastructures. I didn't come to this by any strategic plan that I know of, I've just always had these mental tendencies. My earliest memories (that are clearly not photographs I've re-written as episodic memories) are not words and stories, but are about (or associated with) floorplans and the texture of carpets and furniture.
The idea of "hearing all your thought" seems weird enough to me that I wonder if its a straw man? Or perhaps this is one of those places where people tend to generalize from one example more than they should? I would not naively expect people to be the same in this respect... For reference, I'm mediocre at math, but very verbal according in aptitude testing contexts. I maxed the SAT verbal, participated in (and later coached) policy debate, placed very high in two spelling bees when I was a child, and don't mind playing with morphological rules to make up words that should be in the dictionary, even if they are not. I can juggle reasonably well and am not any worse at social processes than I want to be, given my ethical tendencies.
My usual lack of internal speech doesn't happen because I'm bad with words in general, it is just that I don't find them to be a particularly expressive medium for transmitting most of what is in my head. When I was growing up my family kept piles of scratch paper and pencils in a standard place in the kitchen and living room so that if a conversation became substantive we could sketch our thoughts and use pointing and words and images to communicate what was in our heads. For example, a conversation about gardening might naturally turn into sketching diagrams of vegetable layouts, with 3-D projections and arrows to show how the sun was expected to fall based on nearby trees, and a second diagram to show how the garden would evolve after 3 months as plants grew and died. Those details are hard to discus but easy to diagram.
When playing the game you linked to, I found that sometimes I'd use internal words like "blue red" while I was looking at (for example) the left side of a hole and the bottom side the hole, so that I could "hold the sounds in my audio register" while looking at tiles to see if a particular tile (based on its left and bottom colors) would fit in the hole I had looked at earlier. When I found a matching tile I didn't think "Now I put this in the bottommost opening on the right", instead I'd just move it to the place my kinesthetic memory said that I had previously seen the hole I was trying to fill.
I would bet that the trick of using an audio register to run working memory intensive mental algorithms is probably very very common. Somewhere (unfortunately I forget the cite) I read that that in WWII the near universality of using words for doing math in the language you learned arithmetic was used to suss out spies. Like people who had perfect English/American accents would be asked to talk out long division, and the German spies would fall into muttering "ein, zwei, drei..." and thereby reveal themselves to have been educated in a German speaking school system.
In the tile game, other than using my audio buffer for extra working memory, I found myself either noticing that I had silent stretches where I wasn't making internal speech, but it also seemed that my attention to internal speech was causing me to produce it. Mostly I was trying out phrases to use in this comment to describe what happened in parrallel with actually playing the game. A few times I noticed myself internally commenting on my performance (eg "that was dumb", or "how many bits to uniquely specify a tile" with a rough calculation of four colors x two edges = roughly enough to code for all the tiles) in ways that would lead to optimizations in my playing behavior (by cutting out behavioral tendencies that I could articulately criticize or engaging in optimizations I could verbally suggest). So internal speech wasn't entirely absent, it just wasn't an internal monologue taking the form of "my whole internal mental tableau". It happened mostly in parallel with game performance, rather than narrating the performance.
Now that I think about it, I remember about two years ago I played a non-standard memory game (flipping facedown cards face up two at a time and keeping them iff they are have the same image) and I got a pretty large performance boost from turning them into flashcards for 30 minutes where I counted them as memorized if I could look at the image and connect it to a unique summary phrase like "mohawk kitty" or "fat chair cat" or whatever. The problem is, the language itself might not have been critical there? Perhaps it was just enough image familiarization that visual chunking based on memorized subtle features was easier? I bet an experiment could be designed to test this stuff...
Anyway, hope my data point is helpful :-)
I've been working on "listening" to my pre-thoughts. It's interesting - once you catch it you automatically start translating it into words, or rethinking a thought "audibly"; it's very hard not to. Then I usually make fun of myself by pre-thinking "echo." On the downside, if you don't translate it into words it's vague and harder to remember - thinking with words is awesome.
For more concrete things you can also think about it with some sense - a touch, smell, sound, taste or sight that doesn't need words.
Has anyone developed techniques for thinking without words?
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.
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?
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.
One relevant experience that comes to mind is "doodling" - drawing aimlessly on a piece of paper, which (at least for me) is usually not accompanied by verbal thoughts at all. Yet there is a logic to such doodles; goals coalesce such as filling out a given area or repeating a pattern which are never explicitly formulated. So this is one sense in which "thinking without words" can be part of a commonplace activity.
I've been told that the main impediment to speed-reading is subvocalization - pronouncing the words in your head, as you read them. If you can stop subvocalizing, you're capable of extracting concepts from text much faster.
Learning to speed-read might be a useful step towards your goal. Apply the usual second-hand disclaimers: I haven't tried this myself.
Firstly....I just got here. Poking around. Hi.
With which criteria could we use to determine whether an attempt to 'think without words' was successful? Is there a threshold one could overcome where they could say 'I've done it.....NOW I'm thinking without words"? Wouldn't such a threshold be completely determined subjectively without any sort of verification available? "I think I'm thinking without words".
Also, if I attempt to critically analyze the medium of my own thought process....most likely an impossible thing to do......the characterization that my thoughts are in 'words' is hard to justify. The concepts of mentalese or private language have sought to describe the medium of our thoughts and have had their problems too.
A last thing, the first sentence of your post seems to suggest that language consists solely of words. And we must stop here to clarify that the sobs of a crying person are still part of 'language' even though no 'words' are spoken. Is there a tangible difference between a sob/crying wail and the statement 'I'm sad"? Human language was born from grunts and crude mouth noises. To distinguish between a 'word' and an appropriately understood 'grunt' or vocal sound is highly problematic.
Maybe it would be interesting to ask deaf people how they think. Sign language, written words, purely visual, ...?
Has anyone developed techniques for thinking without words?
Visualisation seems to be the big one.
There is another thing - that I gather helps some people. Sometimes people's ego gets tied into their verbal personality. Then, things that break the hold of their ego over them also help them to break down their conception of themselves as being just the part of them that thinks verbally. A range of traditional techniques aim to produce that result.
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.
Before language, people must have thought without words. I often have the impression that I have a thought fully-formed in my head, yet I wait to listen to it unfold in words before moving on to the next thought. Perhaps I could think much faster if I weren't addicted to words.
Has anyone developed techniques for thinking without words?
This would have a little in common with Buddhist practices of emptying your mind, but wouldn't be the same thing. For one thing, Buddhists also try to empty their minds of images. More importantly, they are trying not to think, while I'm trying to think - just not unpack everything into words.