I'm trying to fathom what it might be like for a person with aphasia and for one with agnosia. For a person with a visual agnosia I wonder this: if two unrecognizable objects are shown to them, and one is entirely within the background of the other, (say a fork on a cutting board) do these two objects mold into a single unrecognizable object or do they remain distinct unrecognizable objects? Or have I got it all wrong and they don't even "see" any object to speak of (regardless of how many or how they are positioned)?
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I think the answer is a qualified yes, humans can think without language. Ultimately though it depends on what you mean by "think."
There is a remarkably large number of deaf people who are not exposed either to a signed or spoken language even in developed countries. These people often invent their own gesture systems, referred to as home sign systems. These people In these cases, the deaf children's systems are more complex than the gestures their parents use (Carrigan & Coppola, 2012). My dad was a home signer until after high school, and he learned how to work on his car, drove a motorcycle, and figured out how the furnace worked. Based on these examples, it is clearly possible to engage in fairly complex reasoning without language. In the article cited below, Spaepen et al. (2011) describe how home signers have some concept of number, albeit imprecise.
This is all to say that thinking is possible without language. Home signers are not catatonic, and can actually engage in complex reasoning. One the other hand, language is a huge cognitive advantage, and in this respect it certainly has an impact on thinking. Language facilitates transmission of information (learning), and it provides a way of organizing thought.
Carrigan, E. M., & Coppola, M. (2012). Mothers Do Not Drive Structure in Adult Homesign Systems: Evidence from Comprehension. In N. Miyake, D. Peebles, & RP Cooper,(Eds.), Proceedings of the 34th Annual Conference of the Cognitive Science Society (pp. 1398-1403).
Spaepen, E., Coppola, M., Spelke, E. S., Carey, S. E., & Goldin-Meadow, S. (2011). Number without a language model. Proceedings of the National Academy of Sciences, 108(8), 3163-3168.
It depends on the type of learning process of a person. One can be better at remembering images, so he might access the images first when trying to define an object/event/etc. This can be thought of as having a photographic memory.
The main reason why most people think with words is because it is a much more sophisticated and structured learning process, and is taught young. Missing the opportunity to learn a language when youre young would most likely cause them to develop a different way of processing information.
Same way goes for someone with visual agnosia. He might develop another way of recognizing objects. For example to use other senses (much like the blind uses his hands to "see"). So one with agnosia would have to use another approach to understand that the fork and the cutting board are two different objects.
Developing (with time or by injury) agnosia or aphasia at some point in your life would certainly mess your usual way of processing information; and it would be very hard to rebuild a new recognition system as old habits are hard to break.
On another note, some friend of mine claims that he consciously thinks with noises when playing and writing music.
Conduct an experiment:
Memorize two texts. It does not matter what kind of texts they are, but you must know them by heart and be able to recite them from memory.
Now start reciting one of those texts (speak it aloud), and at the same time try to think the other text. Or recite one text and write the other, both at the same time.
You will notice that you cannot think of both texts at the same time. You must interrupt your declamation to think or write the other text.
Your finding from this first experiment is that your mind has only the capacity to think one verbal thought at a time. This is important.
Now start reciting one of the texts, and then go to the kitchen and cook a meal, while you keep riciting. Or go to the bathroom and clean it, sink, tub, floor and all.
You will notice, that while your mind is engaged in thinking a verbal thought, you can still plan and execute another activity. And you can perceive the results of your actions, and correct your plan accordingly: if the water cooks, you can lift the lid and put in the pasta without having to stop reciting your text; if your sponge is dry, you can put water and cleaner on it, without having to stop reciting.
This means that you can process information, reflect it and make decistions (i.e. "think") while your verbal processor is occupied.
If you find the examples artificial, consider driving a car while talking on the phone. If you don't use a handheld devide, handling and looking at which will disctact your from driving, but a headset, then you are perfectly able to not run down playing children while carrying on an intense debate with your husband or wife.
But if you can reflect and decide while your verbal memory and thought processes are occupied, it must mean that you think non-verbally.
You have just proven, by experimentation, that human beings can think "without words".
Two articles that summarize expert opinions (Behaviourism, Pinker, Whorf etc.) and attempt an answer to your question. There's more, most of which is not freely accessible on the web. Just search for "thinking without language" on Google Scholar. You can read most of those texts at your local university library.
It is hard to imagine what evidence might be found of thoughts existing without language. Even feral children encounter language (either verbal or nonverbal) through limited contact with people or animals, and lab animals communicate nonverbally with researchers, even if only through their environments (the researcher observes the animal's behavior, and the animal observes how the researcher changes its environment, e.g., its food, waste, and cage furnishings). For reasons such as these, the prospect of thought without language may be somewhat untestable. Producing a sentient being that is never allowed the opportunity to communicate with others would also be unethical by most means I can imagine.
As for your (largely unrelated) question about object discrimination in the absence of recognition, I can't offer any insight in the context of visual agnosia per se, other than to point out that (according to Wikipedia), "Visual agnosia...is not due to a deficit in...language." However, outside that context, I would recommend watching an episode of Scientific American Frontiers, "The Man with Two Brains" that I mentioned in another answer of mine just yesterday! The effects of lacking communication between one's cerebral hemispheres are different from visual agnosia, but give some insight into how unknown objects (e.g., crude drawings) may be identified and differentiated in the absence of linguistic function, which is generally left-hemisphere-dominant. In this episode, Mike Gazzaniga also demonstrates how his split-brain case study subject attends differently to the components that comprise larger objects vs. the overall unity of those objects as one meta-object (coined here for lack of a better term), depending on which hemisphere's visual field processes such an object comprised of smaller objects. The left hemisphere, in which linguistic functions primarily operate, tends to break down these meta-objects and perceive them separately, whereas the right hemisphere tends to notice mostly the overall silhouette of the meta-object. Thus when a still-life painting of a pile of objects that together resemble a face is presented to the right (relatively non-linguistic) hemisphere, a face is perceived, not a bunch of objects (that happen to be organized in a face-resembling pattern).
This sort of evidence might offer the next-best thing to what you'd hope to find through study of visual agnosia. From Gazzaniga's demonstrations with his split-brain case study subject, it seems likely that, without involving the left hemisphere (and thus largely avoiding the involvement of major linguistic processes), the right hemisphere operating alone would perceive a fork on a cutting board as one object in the shape of a cutting board, or would at least require stronger visual contrasts between the two (e.g., color, shadow, texture) to achieve a just-noticeable difference between the two, relative to the left hemisphere operating without the right. Whether that actually has anything to do with language though, or whether it's a separate qualitative difference in the functions of each hemisphere's visual processes, is another question on which I can hardly even begin to speculate.