Visual Thinking / Maria Kozhevnikov Research Re-Post Mental Imagery and Human-Computer Interaction Lab


For those who can’t resist taking brain tests, go to the website. I would love to hear comments by Aspies who look into this research.

Maria Kozhevnikov’s labs at Harvard and NUS jointly investigate the neural mechanisms of visual/spatial imagery, as well as individual differences in basic information processing capacities (e.g. the ability to generate, inspect, or transform visual/spatial images). In addition, the lab research focuses on examining how these individual differences affect more complex activities, such as spatial navigation, learning and problem solving in mathematics and science, as well as in exploring ways to train visual/spatial imagery skills and design learning technologies that can accommodate individual differences and learning styles.

Our research into the object-spatial dissociation follow three directions:

Our central finding is that some individuals use imagery to construct vivid, concrete, and detailed images of individual objects (object visualizers), whereas others use imagery to represent the spatial relationships between objects and perform spatial transformations, such as mental rotation (spatial visualizers). Moreover, our behavioral results showed that there is a trade-off between object and spatial imagery ability while object visualizers score poorly on spatial imagery tasks but excel on object imagery tasks, spatial visualizers score high on spatial imagery tasks  but poorly on object imagery tasks.  

Visualization Ability


“The land, and the land inside me.”

My awareness of the environment is overwhelmingly visual. I have stated before, that if asked “Who are you?” my answer would be, “I am everything that I have ever seen.” When immersed in the surrounding desert, which is mostly devoid of human activity, I experience the landscape intimately – its forms, surfaces and “light” – as if boundaries melt away. This “boundary-less” state probably also explains the Asperger dislike of humanscapes: the sensory “invasion” by human stress, emotions and aggression is highly negative and taxing. The exhaustion that comes with trying to “shut out” chaotic thoughts and constant social strife, simply is too much to endure for very long.
I haven’t explored the questionnaires constructed by Mental Imagery and Human-Computer Interaction Lab, but I would predict that I’m an object visualizer. I have taken spatial tests, but these require “word instructions” beforehand that explain the steps one must follow to solve the puzzle. In other words, (it seems to me) that spatial tests are not entirely visual – they require conscious word activity and “rules” to accomplish the actual geometric tasks.

Musings on THE SELF /


Excerpts various posts:

A child is told who it is, where it belongs, and how to behave, day in and day out, from birth throughout childhood (and indeed, throughout life.) In this way culturally-approved patterns of thought and behavior are imparted, organized and strengthened in the child’s brain. Setting tasks that require following directions (obedience) and asking children to ‘correctly’ answer questions along the way, helps parents and society to discover if the preferred responses are in place.

I don’t remember blurting out “Cogito ergo sum!” in school one day. Achieving awareness of my existence was a misty process, a journey taken before I “knew” of an existence of a “self”. Identity (which is not the same as personality) does not pre-exist; it is constructed. Long before a baby is conceived and born, family and society have composed an identity and a comprehensive world picture for it. The majority of those who belong to a religion or a social class are members by accident of birth, not by choice. We are born into cultures and belief systems; into versions of reality invented by humans long departed.

Self awareness comes as we live our lives: true self-esteem is connected to that process, not as a “before” thing, but an “after” thing: a result of meeting life as it really is, not as a social fantasy. Self awareness is built from the expression of talents and strengths that we didn’t know we possessed. It also arises as we see the “world” as its pretentions crumble before us. Being able to see one’s existence cast against the immensity of reality, and yet to feel secure, is the measure of finally giving birth to a “self”.

As a “hyposocial” individual, tattooing is somewhat of a mystery: tattoos are a social “sign of commitment” to a group or belief system, whether or not that group is large or consists of one other person. My reaction is: But what if you change your mind? What if your “self” changes? The notion of a “static” self is difficult to grasp.

Me, me, me, me, me! The social typical orientation. This is how NTs “look” to me.

One of the big mistakes that social typicals make is to attribute intent to Asperger behavior. This is because social typicals are “self-oriented” – everything is about THEM; any behavior on the part of a human, dog, cat, plant or lifeform in a distant galaxy, must be directed at THEM. Example: God, or Jesus, or whomever, is believed to be paying attention 24/7 to the most excruciatingly trivial moments in the lives of social typicals. We’re not as patient as God or Jesus.

The Asperger default mental state is a type of reverie, day-dreaming, trance or other “reflective” brain process; that is, we do “intuitive” thinking. The “blank face” is because we do not use our faces to do this type of thinking. 

Sorry – we’re just busy elsewhere! When you ask a question, it can take a few moments to “come out of” our “reverie” and reorient our attention. If you are asking a “general question” that is meant to elicit a “feeling” (social) response, it will land like a dead fish in front of us. Hence the continued “blankness”.

The self is “imported” from a socio-cultural menu.

It is a very common assumption that all people “think and act” exactly alike. (Thus the insistence that “underneath it all, everyone is the same” – often said by white people to end discussions of racism) When I was a child I also thought that everyone had “the same brain” as if they roll off an assembly line into our skulls, and it created no end of problems! How could people “come up with” bizarre conclusions and irrational explanations for perfectly logical occurrences? And then one day, I realized that my brain “worked” differently than just about everyone I had ever met. This was a giant leap toward self awareness of the good news / bad news type.   

It is exactly this human self-centeredness that makes the “Theory of Mind” and “mind-reading” so laughable.

Neurotypicals assume that the other person thinks and feels as they do: this is a good “guess” when social people account for 99% of the population and the self is “imported” from an extremely limited socio-cultural menu. And, social people are taught to automatically agree with what others say, in order to be considered a “nice person”. 

Who am I?

The answer for me turned out to be simple: I am everything I have ever seen. Meep! Meep!

especially when young, asks,






Personal thoughts on anxiety in ASD / Asperger Types

My quest is to “untangle” the bizarre mess that “researchers” have created around ASD / Asperger’s symptoms and the “co-morbidity” of anxiety.

How difficult a question is this?

Is anxiety a “big problem” for individuals diagnosed with Asperger’s? If yes, then is it commonly “debilitating” in that it prevents the person from engaging in successful employment, satisfying relationships, and “freedom” to engage the environment by participating in activities that are important to their “happiness”?

And yet, what I encounter are articles, papers, and studies that focus on the argument over whether or not anxiety is part of ASD Asperger’s, the diagnosis, or a co-morbid condition. Anxiety, for “experts” has taken on the “power” of the Gordian knot! Honestly? This is the typical “point” at which an Asperger “looses it” and wants to simply declare that neurotypicals are idiots… but, I’m on a mission to help myself and my co-Aspergerg types to survive in social reality. We’re not going to find logical reality-based “answers” in psychology or even in neuroscience…we are on our own. 

So let’s look at anxiety, another of those words whose meaning and utility have been destroyed by neurotypical addiction to “over-generalization” and fear of specificity!

Over the past few months, I have experienced an increase in “sudden onset” panic attacks: it’s not as if I can’t assign a probable cause. The facts of my existence (age, health, financial problems) are enough to fill up and overflow whatever limit of tolerance that I can summon up each day. Severe (and sometimes debilitating) anxiety has been integral to my existence since at least age 3, which is the time of my first “remembered” meltdown. I can honestly say, that if it were not for “anxiety” manifesting as sudden meltdowns, panic attacks, “background radiation” and other physical  reactions, (who cares what they are labeled?), my life would have been far easier, with much more of my time and energy being available to “invest” in activities of choice, rather than surviving the unpredictable disruptions that I’ve had to work around. The fact that I’ve had an interesting, rich and “novel” existence, is thanks to maximizing the stable intervals between anxiety, distress, and exhaustion – and avoiding alien neurotypical social expectations and toxic environments as much as possible.

Here is a simple formula that I have followed:

Life among NTs is HELL. I deserve to “reserve” as much time as possible for my intrinsically satisfying interests; for pursuit of knowledge, experiences and activities that enable me to become as “authentic” to “whoever and whatever I am” as possible.

This realization came long, long before diagnosis, and I had to accept that a distinct possibility was that there was no “authentic me” and if there was, it might be a scary discovery. But, ever-present Asperger curiosity and dogged persistence would accept no other journey. It is important to realize, that Asperger or not, this type of “classic quest” has been going on in human lives for thousands of years, and for the most part has been in defiance of social disapproval (often regarded as a serious threat) by societies world-wide, which impose on individuals the carefully constructed catalogue of roles and biographies handed down from “on high”.

The point is that the choice to “go my own way” was “asking for it” – IT being endless shit (and the accompanying anxiety) dumped on human beings existing on all levels of the Social Pyramid, but especially directed toward any group or individual who is judged to be “antisocial” or inferior. I have encountered conflicts large and small, and was exposed to “human behavior” in ways I couldn’t have imagined.

What I have confronted in “normdom” is the strange orientation of “experts” who ignore the contribution of environmental sources to hyperarousal, a physiological reaction to conditions in the environment. (Note: Fear, anxiety, and all the “emotion-words”  are merely the conscious verbal expression that infants and children ARE TAUGHT to utilize in social communication, and for social purposes) These words are not the physiological experience.

A feedback “loop” exists between the environment and the human sensory system.   The physiology of fear and anxiety is an ancient “alarm system” that promotes survival, but in the human behavior industry, anxiety has been “segregated” and  classified as a pathology – an utterly bizarre, irrational, and dangerous idea. The result is that “normal” human reactions and behavior, provided by millions of years of evolutionary processes, and which  PROTECT the individual, are now “forbidden” as “defects” in the organism itself. Social involvement and culpability are “denied” – responsibility for abuse of humans and animals by social activity is erased!

Social indoctrination: the use of media, advertising, marketing, political BS and constant “messaging” that presents “protective evolutionary alerts and reactions” (awareness of danger; physiological discomfort, stress and illness) are YOUR FAULT. You have a defective brain. It’s a lie.

Due to an entrenched system of social hierarchy (inequality), social humans continue to be determined to “wipe out” the human animal that evolved in nature, and replace it with a domesticated / manufactured / altered Homo sapiens that just like domesticated animals, will survive and reproduce in the most extreme and abusive conditions.

This “domestic” hypersocial human is today represented as the pinnacle of evolution.

Human predators (the 1 %  who occupy “power positions” at the top of the pyramid)merely want to ensure that the status quo is maintained, that is, the continued  exploitation of the  “observation” that domesticated humans will adapt to any abuse – and still serve the hierarchy. This “idea” also allows for the unconscionable torture and abuse of animals.

The “expert” assumption is that a normal, typical, socially desirable human, as defined by the “human behavior” priesthood, can endure any type and degree of torture, stress, abuse, both chronic or episodic, and come out of the experience UNCHANGED; undamaged and exploitable. Any variation from this behavioral prescription is proof of a person’s deviance, inferiority and weakness.

The most blatant example of this “attitude” is the epidemic of PTSD and suicide in soldiers returning from HELL in combat. Not that many wars ago, militaries literally “executed”  soldiers suffering from this “weakness, cowardice and treason” on the battlefield, or “exiled” them to asylums as subhuman and defective ‘mistakes”. Now we ship soldiers home who have suffered extreme trauma and “treat them” so badly, that suicide has become the only relief for many. Having the afflicted remove him or herself, rather than “murdering” them is considered to be compassionate progress.  

And my point is about relief: I concluded long ago that chronic and episodic “hyperarousal” must be treated immediately with whatever works; in my experience, that means medication. Despite limiting one’s “exposure” to toxic social environments, one cannot escape the damage done to human health and sanity.

Some relief can be had by employing activities and adjustments in thinking patterns, that often (usually by trial and error) can mitigate physical damage. But what we must remember is that anxiety, fear, distress and the “urge to flee” are healthy responses to horrible human environments. How many mass migrations of “refugees” are there at any time, with thousands, and even millions of people, seeking “new places” to live a life that is proper to a healthy human?




The Amygdala Is NOT the Brain’s Fear Center

Separating findings from conclusions

by Joseph LeDoux, Ph.D., directs the Emotional Brain Institute at NYU and at the Nathan Kline Institute for Psychiatric Research. He is author of Anxious: Using the Brain to Understand and Treat Fear and Anxiety.


I’ve been studying the amygdala for more than 30 years. When I started this work, research on this brain region was a lonely field of inquiry. The hippocampus was all the rage, and I sometimes felt jealous of the attention lavished on this brain region because of its contribution to memory.  These days, though, it is the amygdala that is in the spotlight.  This little neural nugget has gone from an obscure area of the brain to practically a household word, one that has come to be synonymous with “fear.” And for many people, my name, too, is practically synonymous with “fear.”  I am often said to have identified the amygdala as the brain’s “fear” center.  But the fact is, I have not done this, nor has anyone else. The idea that the amygdala is the home of fear in the brain is just that—an idea. It is not a scientific finding but instead a conclusion based on an interpretation of a finding.  So what is the finding, what is the interpretation, and how did the interpretation come about?

The Finding:  When the amygdala is damaged, previously threatening stimuli come to be treated as benign.  The classic discovery was that monkeys with amygdala damage were “tamed;” snakes, for example, no longer elicited so-called fight-flight responses after amygdala damage.  Later studies in rats by me, and others, mapped out the amygdala’s role in a neural system that detects and responds to threats, and similar circuits were found to be operative when the human brain processes threats.

The Interpretation: Since damage to the amygdala eliminates behavioral responses to threats, feelings of “fear” are products of the amygdala. People are indeed less responsive to threats when the amygdala is damaged (in humans amygdala damage can occur as a result of epilepsy or other medical conditions or their surgical treatment). Yet, these people can still experience (feel) “fear.” In other words, the amygdala is an important part of the circuit that allows the brain to detect and respond to threats, but is not necessary to feel “fear.”

Brain imaging studies of healthy humans (people without brain damage) suggest something similar. When they are exposed to threats, neural activity in the amygdala increases and body responses (like sweating or increased heart rate) result. This is true even if the threatening stimuli are presented subliminally, such that the person is not consciously aware that the threat is present and does not consciously experience (feel) “fear.”  Amygdala activity does not mean that fear is experienced.

The conclusion that the amygdala is the brain’s fear center wrongly assumes that the feelings of “fear” and the responses elicited by threats are products of the same brain system. While amygdala circuits are directly responsible for behavioral/physiological responses elicited by threats, they are not directly responsible for feelings of “fear.”

How did the interpretation come to be?  We humans frequently feel afraid when we find ourselves freezing or fleeing when in harm’s way. In other words, these two things (the feeling and the body responses) tend to be tightly correlated in our conscious introspections. (The verbal “version” of what happened) These introspections are talked about and become shared experiences that are ingrained as natural truths. Most people thus believe that the feeling of fear is the reason an animal or person runs from danger; or that the classic facial expression we know as “fear” is driven by feeling afraid.  But when it comes to the brain, what is obvious is not always what is the case. The purpose of science is to go beyond the obvious to reveal the deeper truths that cannot be gleaned simply from observing nature.

Are fear and the myriad other “emotions” learned ? 

One of the first things a scientist learns is that a correlation does not necessarily reveal causation.  (Not really; this mistake is the bread and butter of psychology) The interpretation that the amygdala is the brain’s fear center confuses correlation and causation.

Actually, there are two confusions involved: (1) because we often feel afraid when we are responding to danger, (we assume that) fear is the reason we respond the way we do; and (2) because the amygdala is responsible for the response to danger, (we conclude that) it must also be responsible for the feeling of fear.

From the beginning, my research suggested that the amygdala contributes to non-conscious aspects of fear, by which I meant the detection of threats and the control of body responses that help cope with the threat. Conscious fear, I argued in my books The Emotional Brain (Simon and Schuster, 1996) and Synaptic Self (Viking, 2002), and most recently in Anxious (Viking, 2015), is a product of cognitive systems in the neocortex that operate in parallel with the amygdala circuit.  But that subtlety – (the distinction between conscious and non-conscious aspects of fear) – was lost on most people.

When one hears the word “fear,” the pull of the vernacular meaning is so strong that the mind is compelled to think of the feeling of being afraid.  For this reason, I eventually concluded that it is not helpful to talk about conscious and non-conscious aspects of fear.  A feeling like “fear” is a conscious experience. To use the word “fear” in any other way only leads to confusion.

The amygdala has a role in fear, but it is not the one that is popularly described. It’s role in fear is more fundamental and also more mundane.  It is responsible for detecting and responding to threats, and only contributes to feelings of fear indirectly.  For example, the amygdala outputs driven by threat detection alter information processing in diverse regions of the brain.  One important set of outputs result in the secretion of chemicals throughout the brain (norepinephrine, acetylcholine, dopamine, serotonin) and body (hormones such as adrenalin and cortisol).  In situations of danger, these chemicals alert the organism that something important is happening. As a result, attention systems in the neocortex guide the perceptual search of the environment for an explanation for the highly aroused state.  The meaning of the environmental stimuli present is added by the retrieval of memories. If the stimuli are known sources of danger, “fear” schema are retrieved from memory.  My hypothesis, then, is that the feeling of “fear” results when the outcome of these various processes (attention, perception, memory, arousal) coalesce in consciousness and compel one to feel “fear.” This can only happen in a brain that has the cognitive wherewithal (to) have the concept of “me,” or what Endel Tulving has called “autonoetic consciousness.”  In a later post, I will elaborate on the autonoetic nature of our conscious feelings. (I’m confused: Does this mean that (IF) no animal other than Homo sapiens experiences self-awareness, or some similar “ME- NESS” then Homo sapiens is the only animal that suffers the “feeling” of fear? Lucky “lower” animals…) 

There’s nothing wrong with speculation in science (I just speculated about how feelings come about). But when a speculative interpretation becomes ingrained in the culture of science, and the culture at large, as an unquestioned fact, we have a problem. 

This problem is especially acute in neuroscience, where we start from mental state words (like fear) that have historical meanings, and treat the words as if they are entities that live in brain areas (like the amygdala).

Yes, yes, yes. This is EXACTLY what drives me nuts. 

In sum, there is no fear center out of which effuses the feeling of being afraid. “Fear” is, in my view, better thought of as a cognitively assembled conscious experience that is related to threat processing, but that should not be confused with the non-conscious processes that detect and control responses to threats.

Postscript:  Be suspicious of any statement that says a brain area is a center responsible for some function.

The notion of functions being products of brain areas or centers is left over from the days when most evidence about brain function was based on the effects of brain lesions localized to specific areas.

Psychology is stuck in the Old Days; the entire basis for  Asperger’s individuals being labeled as pathological and developmentally defective is nonsense. 

Today, we think of functions as products of systems rather than of areas. Neurons in areas contribute because they are part of a system. The amygdala, for example, contributes to threat detection because it is part of a threat detection system.  And just because the amygdala contributes to threat detection does not mean that threat detection is the only function to which it contributes. Amygdala neurons, for example, are also components of systems that process the significance of stimuli related to eating, drinking, sex, and addictive drugs.

Psychology of Pro Hockey Players / Useful Tips for Asperger Types

From Sports Psychology Today 

It’s that time of year: I’m a huge hockey fan, having grown up with the Chicago Blackhawks on black and white TV. No helmets. Scarred faces, missing teeth. Much less padding. But were there more fights than today? I honestly can’t say.

Today it’s (my friend’s) vast screen TV and obnoxious pregame “shows” (what else can be expected from Las Vegas, that great faux-gold turd in the desert). But I wouldn’t want to deprive anyone from experiencing the “joy of” the Stanley Cup Final. My friend has never been a fan of hockey (or any sport), but enforced viewing through some of the playoff games, and she’s “hooked” already – unfortunately, by the Las Vegas Knights. My guess concerning my friend’s rapid “seduction” by hockey? Well it is a fantastic sport, after all! And a chance for female Homo sapiens to observe and enjoy “men being men” without any collateral damage, like in war. They can punch each other all they want; no one else gets hurt. 

I have also observed that the “mental qualities” manifested by players may be a guide to help nervous Asperger types approach our confrontations with “hostile” neurotypicals. No kidding!  

Performance Anxiety and Pregame Jitters

Written by

Many athletes feel performance anxiety in the opening minutes of the game. You may feel butterflies in your stomach or your heart pounding. Some athletes like to feel pregame jitters before competition. These athletes think of pregame jitters as a sign of readiness and energy. Other athletes think of pregame jitters as a sign of nervousness.

I would say that “pregame jitters” are a fact of life for Asperger types: every social interaction, everyday – along with a strong tendency to “rehearse” upcoming events – but aren’t daily practice and visualization vital to athletic excellence? Can we change our “attitude” toward this anticipatory physical phenomenon, and perhaps take a “neutral” view? I know, it seems a difficult task! LOL

Pre-game jitters are a natural part of competing and a sign you are ready to embrace competition. Even the best athletes in the world get the jitters.   Michael Leighton, goaltender for the Philadelphia Flyers, admitted to feeling nervous before his first NHL playoff game.”My legs were shaking a little bit, I was nervous,” Leighton said. “Once I made a few saves, you kind of forget about that and just get focused. It kind of goes away.”

This seems applicable to Asperger types; often, once I get to the “performance” part of social interaction, something automatic takes over and I jabber away – 

The mistake many athletes make is interpreting pre-game jitters as there is something wrong or a problem.  Pregame jitters can be harmful when they don’t go away in the opening minutes of the game. They can cause you to lose confidence and focus. When you’re focused on how nervous you feel, you lose focus on the present task.

Athletes need to embrace the pregame jitters as a sign they are ready to play.  Your mental game tip is to stay calm when you experience pregame jitters in the opening minutes. (Yes, but how???) Stay focused on your strategy and what’s important to execute.  Pregame jitters are important to help you prepare for the game and they will help you focus your best if you embrace them! Think of it this way: the best athletes get worried if they don’t experience pregame jitters!

Maybe our tendency to rehearse is an asset, if we use that energy to devise a strategy! Is rehearsal another “asset” that NT psychologists misinterpret as a defect?

Listed below are some mental game tips to help you perform your best under pressure and in the big game.

What seem like minor everyday social interactions for NTs can be extremely “big game” status for us! 

1. Develop a consistent pregame routine. (Yes, psychologists judge preference for establishing routines in ASD / Asperger individuals a “developmental defect”. Screw them! We need to use our traits as assets…)  A pregame routine can help transition you into the right mindset before competition. While you’re warming up your body, you also want to mentally prepare for the upcoming game. A pregame routine will help you focus your mind, prepare to feel confident, and to trust in your practice. During your pregame routine, remind yourself to trust in the practice you have done leading up to the game.

2. Focus on your game not your competitors. Many athletes tend to make comparisons to their competitors and thus psych themselves out. When you do this, you typically make negative comparisons, which can cause you to lose confidence in your game. Instead of gawking at your competitors, you want to focus on your pregame preparation. You should focus on your strengths and abilities, for example.

3. Focus on the process, not results. Focusing on results causes you to think too far ahead and sets too many expectations for competition. When you focus on the results, you lose focus on the current play, point or shot and you can’t perform your best. Remind yourself that focusing on results doesn’t help you execute. Then, refocus quickly on what’s important, such as the target or your strategy for the current play.

4. Have trust in yourself. Some athletes lose trust and tighten up in the big game. This can cause you to over control your performance and not play freely. You want your performance to just happen, without thinking too much about “how to” execute your skills. For example, a batter needs to react to the ball instead of think about how to make a good swing. Simplify your thoughts such as thinking about one thought or image to help you execute (feeling balanced). Avoid thinking too much about how to or technique.

Overall, you want to treat the big game as any other game. You don’t want to place too much importance on one game, which can lead to added pressure, a lack of focus, and trust in your game. Focus on what you do best. Follow your usual pregame routine and mentally prepare for the big game just like you would any other game.

What does all this point to for Asperger types? We must free ourselves from the poisonous messages we have received all our lives from neurotypicals who “judge” our traits and behaviors as “defective and subhuman” And USE our cognitive skills and superior sensory abilities to our advantage! This is very different than submitting to being “trained monkeys” as social humans demand of us. 


Contemplating Dream Experiences

Where did the world go?

Where did the world go?

After I woke up from a particularly confusing jumble of dream images one morning, it occurred to me that the brain during sleep may be reacting to being cut off from the environment, as if it’s locked in a dark closet. The brain depends on a stream of information arriving from the senses; it uses this information to make sense of the environment and to model “reality.” Maybe it ‘freaks out’ when the visual information stream shuts down.

During REM sleep the brain tries to combine peripheral sensations with memory (like sounds from the street, or temperature changes in the room) but without the necessary full connection to the “outer world” via the senses, a coherent story can’t be composed. That is, the brain’s function, which is to make sense of the environment, is  to write an ongoing story that integrates all the available information that the brain needs to direct and control the body.

Deprived during sleep of sense information, especially visual orientation, the brain simply can’t thread images, sounds and motion into a coherent story. Whatever we may “dream” it is mostly forgotten, and if we do remember, the brain then strives to make a reasonable story from fragments that we can recall.

A Plea for Visual Thinking / Rudolf Arnheim

A Plea for Visual Thinking

see also an interview with RA:

Rudolf Arnheim Reviewed work(s): Source: Critical Inquiry, Vol. 6, No. 3 (Spring, 1980), pp. 489-497 Published by: The University of Chicago Press Stable URL: . Accessed: 31/01/2013 13:04

Perception and thinking are treated by textbooks of psychology in separate chapters. The senses are said to gather information about the outer world; thinking is said to process that information. Thinking emerges from this approach as the “higher,” more respectable function, to which consequently education assigns most of the school hours and most of the credit. The exercise of the senses is a mere recreation, relegated to spare time.

It is left to the playful practice of the arts and music and is readily dispensed with when a tight budget calls for economy. The habit of separating the intuitive from the abstractive functions, as they were called in the Middle Ages, goes far back in our tradition. Descartes, in the sixth Meditation, defined man as “a thing that thinks,” to which reasoning came naturally (it obviously doesn’t!); whereas imagining, the activity of the senses, required a special effort and was in no way necessary to the human nature or essence. (The arts and technology are vital to human health and happiness -)

Note: We see the “elevation” of these narrow ideas about “a hierarchy of thinking” (that damn pyramid obsession again) in the denigration of ASD / Asperger abilities: (formal, old-fashioned use of language if language is present; echoing or copying (parroting) of language with an extensive “memorized” vocabulary, but without a “clue” to the “deeper meaning”  of language; an indictment of ASD / AS individuals as robots that are utterly lacking in imagination or creativity; as enthralled by boring subject matter (to social types) and above all, the failure to accomplish what has recently been elevated to the “highest level of cognition attainable, socio-emotional language, exemplified by: Have a nice day!

For “verbally deficient” autistics, this means an immediate judgement of low intelligence.  

So far, we have a very clear historical explanation as to why “visual-sensory thinking” got trashed, demoted and eventually designated as a “developmental disability” by American psychologists. This vital and creative cognitive process has vanished from the “acceptable human social repertoire” of “brain activity” in puritanical” American culture.  

The passive ability to receive images of sensory things, said Descartes, would be useless if there did not exist in the mind a further and higher active faculty capable of shaping these images and of correcting the errors that derive from sensory experience. (Exactly backwards to how thinking works) A century later Leibniz spoke of two levels of clear cognition.’ Reasoning was cognition of the higher degree: it was distinct, that is, it could analyze things into their components. Sensory experience, on the other hand, was cognition of the lower order: it also could be clear but it was confused, in the original Latin sense of the term; that is, all elements fused and mingled together in an indivisible whole. Thus artists, who rely on this inferior faculty (as do many top inventors and scientists), are good judges of works of art but when asked what is wrong with a particular piece that displeases them can only reply that it lacks nescio quid, a certain “I don’t know what.” (Intuitively, you “get it” or you don’t)

Yes, the Descartes – thing is nonsense. Just because a man is a genius is one field, doesn’t mean that he is an expert on everything; but NTs love authority and will believe without question what “great men” say. Our present predicament of relying on a “false pyramid of thinking” based on “dumb” (not reasonable) value judgements from (European white male) heroes of the past, has devastated the power of thinking “outside the box of verbal abstraction and generalities” in entire societies.

In our own time, language has been designated as the place of refuge from the problems incurred in direct perceptual experience; this in spite of the fact that language, although a powerful help to our thinking, does not offer in and by itself an arena in which thinking can take place. Thus the very title of a recent collection of articles by Jerome S. Bruner suggests that in order to arrive at knowledge the human mind must go “beyond the information given” by direct sensory experience. Bruner adopts the belief that the cognitive development of a child passes through three stages. The child explores the world first through action, then through imagery, and finally through language. 

The implication is, unfortunately, that with the arrival at a next level the earlier one falls by the wayside.

This is obviously untrue: adults retain modes of “thinking” from childhood stages. Magical thinking is the default mode of thinking for neotenic social typicals. Magic  “fills in” the gaps left by inferior sensory data and perception, supplying “fantastical” explanations for phenomena. Reasoning, critical analysis, and effective understanding of “how the universe works” (math-science) may be native to a few individuals, but must be taught and cultivated in the majority of children. This is a taboo in highly religious American culture. Reality-based thinking has been abandoned, even demonized, in American education – and for several generations – in favor of socially-promoted emotional narcissism that contributes to a very distorted social reality and description of “being human.” That is, a supernatural orientation is the result of developmental stagnation, and furnishes the status quo in religious, psychological and social engineering regimes. Neoteny is a fact of life for the modern social human. 

Thus when the child learns to go beyond a particular constellation directly given to his eyes, the ability to restructure the situation in a more suitable way is not credited by Bruner to the maturing of perceptual capacity but to the switch toward a new processing medium, namely, language. Thus language is praised as the indispensable instrument for essential refinements of the mind, toward which in fact, language is little more than a reflector.

To claim that “cognition” suddenly appeared out of nowhere, only with the “arrival of human verbal language” is idiotic and unbelievably arrogant! 550 million years of “arms race” evolution, but “sensory thinking” is inferior…

We are told by psychologists that “autistic” children are defective (low intelligence) due to two outrageous prejudices:

1. Lack of verbal language use, and/or failure to use language as prescribed (social scripts) is automatically a “sign” of defective development. (This overturns and discards 550 millions of years of evolution)

2. Superior sensory perception and processing, which are autistic strengths, are denigrated as ‘low-level’ cognition.

Since experts insist that perception offers nothing better than the fairly mechanical recording of the stimuli arriving at the sensory receptors, it is useful to respond with a few examples which show that perception transcends constantly and routinely the mere mechanical recording of sensory raw material. (I am limiting myself in the following to visual perception.) At a fairly simple level, the psychologist Roger N. Shepard and his coworkers have shown that visual imagination can rotate the spatial position of a given object when a different view is needed to solve a problem, for example, in order to identify the object with, or distinguish it from, a similar one. (I have noted previously that this type of “test” is a very limited and rule-based conception of what visual thinking can and does accomplish) This is worth knowing. But reports by artists and scientists indicate that visual imagination is capable of much more spectacular exploits. Indeed, the imagination of the average person demands our respect.

Let me use an example cited in an article by Lewis E. Walkup. The solution of the puzzle should be attempted without the help of an illustration. Imagine a large cube made up of twenty-seven smaller cubes, that is, three layers of nine cubes each. Imagine further that the entire outer surface of the large cube is painted red and ask yourself how many of the smaller cubes will be red on three sides, two sides, one side, or no side at all.

SEE Skipped TEXT

Far from abandoning our image, we discovered it to be a beautiful, composition, in which each element was defined by its place in the whole. Did we need language to perform this operation? Not at all; although language could help us to codify our results. Did we need intelligence, inventiveness, creative discovery? Yes, some. In a modest way, the operation we performed is of the stuff that good science and good art are made of.

Was it seeing or was it thinking that solved the problem? Obviously, the distinction is absurd.

In order to see we had to think; and we had nothing to think about if we were not looking. But our claim goes farther. We assert not only that perceptual problems can be solved by perceptual operations but that productive thinking solves any kind of problem in the perceptual realm because there exists no other arena in which true thinking can take place. Therefore it is now necessary to show, at least sketchily, how one goes about solving a highly “abstract” problem. For the sake of an example, let me ask the old question of whether free will is compatible with determinism. Instead of looking up the answer in Saint Augustine or Spinoza, I watch what happens when I begin to think. In what medium does the thinking take place? Images start to form. Motivational forces, in order to become manipulable, take the shape of arrows. These arrows line up in a sequence, each pushing the next-a deterministic chain that does not seem to leave room for any freedom (fig. la). Next I ask What is freedom? and I see a sheaf of vectors issuing from a base (fig. lb). Each arrow is free, within the limits of the constellation, to move in any direction it pleases and to reach as far as it can and will. But there is something incomplete about this image of freedom. It operates in empty space, and there is no sense to freedom without the context of the world to which it applies. My next image adds an external system of a world minding its own business and thereby frustrating the arrows that issue from my freedom-seeking creature (fig. ic). I must ask: Are the two systems incompatible in principle? In my … GO TO:


ASD / AS Intelligence Revisited / Guess what? We’re intelligent. DUH!

PLoS One. 2011; 6(9): e25372.
Published online 2011 Sep 28. doi:  10.1371/journal.pone.0025372
PMID: 21991394

The Level and Nature of Autistic Intelligence II: What about Asperger Syndrome?

Isabelle Soulières, 1 , 2 , * Michelle Dawson, 1 Morton Ann Gernsbacher, 3 and Laurent Mottron  / Efthimios M. C. Skoulakis, Editor


Individuals on the autistic spectrum are currently identified according to overt atypicalities in socio-communicative interactions, focused interests and repetitive behaviors [1]. More fundamentally, individuals on the autistic spectrum are characterized by atypical information processing across domains (social, non-social, language) and modalities (auditory, visual), raising the question of how best to assess and understand these individuals’ intellectual abilities. Early descriptions [2], [3] and quantifications (e.g. [4]) of their intelligence emphasized the distinctive unevenness of their abilities. While their unusual profile of performance on popular intelligence test batteries remains a durable empirical finding [5], it is eclipsed by a wide range of speculative deficit-based interpretations. (based on socio-cultural arrogance) Findings of strong performance on specific tests have been regarded as aberrant islets of ability arising from an array of speculated deficits (e.g., “weak central coherence”; [6]) and as incompatible with genuine human intelligence.

For example, Hobson ([7], p. 211) concluded that regardless of strong measured abilities in some areas, autistics lack “both the grounding and the mental flexibility for intelligent thought.

Thus, there is a long-standing assumption that a vast majority of autistic individuals are intellectually impaired. In recent years, this assumption has been challenged by investigations that exploit two divergent approaches —represented by Wechsler scales of intelligence and Raven’s Progressive Matrices— to measuring human intelligence [8]. Wechsler scales estimate IQ through batteries of ten or more different subtests, each of which involves different specific oral instructions and tests different specific skills. The subtests are chosen to produce scores that, for the typical population, are correlated and combine to reflect a general underlying ability. Advantages of this approach include the availability of subtest profiles of specific skill strengths and weaknesses, index scores combining related subtests, and dichotomized Performance versus Verbal IQ scores (PIQ vs. VIQ), as well as a Full-Scale IQ (FSIQ) score. However, the range of specific skills assayed by Wechsler scales is limited (e.g., reading abilities are not included), and atypical individuals who lack specific skills (e.g., typical speech processing or speech production) or experiences (e.g., typical range of interests) may produce scores that do not reflect those individuals’ general intelligence.

In contrast, Raven’s Progressive Matrices (RPM) is a single self-paced test that minimizes spoken instruction and obviates speech production or typicality of experiences [9]. The format is a matrix of geometric designs in which the final missing piece must be selected from among an array of displayed choices. Sixty items are divided into five sets that increase progressively in difficulty and complexity, from simple figural to complex analytic items. RPM is regarded both as the most complex and general single test of intelligence [10], [11] and as the best marker for fluid intelligence, which in turn encompasses reasoning and novel problem-solving abilities [8], [12]. RPM tests flexible co-ordination of attentional control, working memory, rule inference and integration, high-level abstraction, and goal-hierarchy management [13], . These abilities, as well as fluid intelligence itself, have been proposed as areas of deficit in autistic persons, particularly when demands increase in complexity [16], [17], [18], [19].

Against these assumptions, we reported that autistic children and adults, with Wechsler FSIQ ranging from 40 to 125, score an average 30 percentile points higher on RPM than on Wechsler scales, while typical individuals do not display this discrepancy, as shown in Figure 1 [20]. RPM item difficulty, as reflected in per-item error rate, was highly correlated between the autistic and non-autistic children (r = .96). An RPM advantage for autistic individuals has been reported in diverse samples. Bolte et al. [21] tested autistic, other atypical (non-autism diagnoses), and typical participants who varied widely in their age and the version of Wechsler and RPM they were administered; autistics with Wechsler FSIQ under 85 were unique in having a relative advantage on RPM. Charman et al. [22] reported significantly higher RPM than Wechsler scores (FSIQ and PIQ) for a large population-based sample of school-aged autistic spectrum children. In Morsanyi and Holyoak [23], autistic children, who were matched with non-autistic controls on two Wechsler subtests (Block Design and Vocabulary), displayed a numeric, though not significant, advantage within the first set of Raven’s Advanced Progressive Matrices items.

The nature of autistic intelligence was also investigated in an fMRI study [24]. Autistics and non-autistics matched on Wechsler FSIQ were equally accurate in solving the 60 RPM items presented in random order, but autistics performed dramatically faster than their controls. This advantage, which was not found in a simple perceptual control task, ranged from 23% for easier RPM items to 42% for complex analytic RPM items.

Autistics’ RPM task performance was associated with greater recruitment of extrastriate areas and lesser recruitment of lateral prefrontal and medial posterior parietal cortex, illustrating their hallmark enhanced perception [25].

One replicated manifestation of autistics’ enhanced perception is superior performance on the Wechsler Block Design subtest, suggesting a visuospatial peak of ability [26]. Even when autistics’ scores on all other Wechsler subtests fall below their RPM scores, their Block Design and RPM scores lie at an equivalent level [20].

Thus, enhanced occipital activity, superior behavioral performance on RPM, and visuospatial peaks co-occur in individuals whose specific diagnosis is autism, suggesting an increased and more autonomous role of perception in autistic reasoning and intelligence [24].

But what about individuals whose specific diagnosis is Asperger syndrome? In Dawson et al.’s previous investigations of autistics’ RPM performance, Asperger individuals were excluded. Asperger syndrome is a relatively low-prevalence [27] autistic spectrum diagnosis characterized by intelligence scores within the normal range (non-Asperger autistics may have IQs in any range). Two main distinctions between the specific diagnosis of autism and Asperger syndrome are relevant to the question of intelligence in the autistic spectrum. First, while their verbal and nonverbal communication is not necessarily typical across development, Asperger individuals do not, by diagnostic definition, exhibit characteristic autistic delays and anomalies in spoken language. While both autistic and Asperger individuals produce an uneven profile on Wechsler subtests, Asperger individuals’ main strengths, in contrast with those of autistics (see [20]), are usually seen in verbal subtests (count me in)  (as illustrated in Figure 2; see also [28]). Although RPM is often deemed a “nonverbal” test of intelligence, in practice typical individuals often rely on verbal abilities to perform most RPM items. (NOTE: I have commented on this in another post, regarding the pre-test tutoring available to students, during which the “rules of the game” are explained. Is this “cheating” in that “fluid intelligence” and not learned procedures, are supposedly being measured?)  

Second, at a group level, Asperger individuals do not display the autistic visuospatial peak in Wechsler scales; rather, their Block Design subtest performance tends to be unremarkably equivalent to their FSIQ (see Figure 2 and also [32]). The question of whether Asperger individuals display the autistic advantage on RPM over Wechsler is thus accompanied by the possibility that the Asperger subgroup represents an avenue for further investigating the nature of this discrepancy. (I am quite baffled at times by my “native” Asperger experience, which is overwhelmingly visual-sensory, but that verbal language is a “go to tool” for translating that experience into “acceptable” form. Very practical! Why does this “arrangement” seem to occur in Asperger’s?)

Our goal was to investigate whether the autistic advantage on RPM is also characteristic of Asperger syndrome and, further, whether RPM performance reveals a fundamental property of intelligence across the autistic spectrum. If the mechanism underlying autistics’ advantage on RPM is limited to visuospatial peaks or to language difficulties disproportionately hampering Wechsler performance, then the advantage should not be found in Asperger individuals. Indeed, as predicted by Bolte et al. [21], Asperger individuals should perform even better on Wechsler scales than on RPM. If instead the underlying mechanism is more general and versatile, then Asperger individuals should demonstrate at least some advantage on RPM. Preliminary findings have suggested this to be the case. In one recent study, Asperger children (age 6–12) obtained significantly higher raw scores on RPM than did typical children matched on age and Wechsler performance [33].

For all the “poo-bah” and graphs, go to original paper (and related papers):


Asperger individuals differ from autistics in their early speech development, in having Wechsler scores in the normal range, and in being less likely to be characterized by visuospatial peaks. In this study, Asperger individuals presented with some significant advantages, and no disadvantages, on RPM compared to Wechsler FSIQ, PIQ, and VIQ. Asperger adults demonstrated a significant advantage, relative to their controls, in their RPM scores over their Wechsler FSIQ and PIQ scores, while for Asperger children this advantage was found for their PIQ scores. For both Asperger adults and children and strikingly similar to autistics in a previous study [20], their best Wechsler performances were similar in level to, and therefore plausibly representative of, their general intelligence as measured by RPM.

We have proposed that autistics’ cognitive processes function in an atypically independent way, leading to “parallel, non-strategic integration of patterns across multiple levels and scales” [36] and to versatility in cognitive processing [26].

Such “independent thinking” suggests ways in which apparently specific or isolated abilities can co-exist with atypical but flexible, creative, and complex achievements. Across a wide range of tasks, including or perhaps

especially in complex tasks, autistics do not experience to the same extent the typical loss or distortion of information that characterizes non-autistics’ mandatory hierarchies of processing

Therefore, autistics can maintain more veridical representations (e.g. representations closer to the actual information present in the environment) when performing high level, complex tasks. The current results suggest that such a mechanism is also present in Asperger syndrome and therefore represents a commonality across the autistic spectrum. Given the opportunity, different subgroups of autistics may advantageously apply more independent thinking to different available aspects of information: verbal information, by persons whose specific diagnosis is Asperger’s, and perceptual information, by persons whose specific diagnosis is autism.

One could alternatively suggest that the construct measured by RPM is relative and thus would reflect processes other than intelligence in autistic spectrum individuals. However, a very high item difficulty correlation is observed between autistic individuals and typical controls, as well as between Asperger individuals and typical controls. As previously noted [20], these high correlations indicate that RPM is measuring the same construct in autistics and non-autistics, a finding now extended to Asperger syndrome.

Therefore, dismissing these RPM findings as not reflecting genuine human intelligence in autistic and Asperger individuals would have the same effect for non-autistic individuals.

The discrepancies here revealed between alternative measures of intelligence in a subgroup of individuals underline the ambiguous non-monolithic definition of intelligence. Undoubtedly, autistics’ intelligence is atypical and may not be as easily assessed and revealed with standard instruments. But given the essential and unique role that RPM has long held in defining general and fluid intelligence (e.g., [37]),

we again suggest that both the level and nature of autistic intelligence have been underestimated.

Thus, while there has been a long tradition of pursuing speculated autistic deficits, it is important to consider the possibility of strength-based mechanisms as underlying autistics’ atypical but genuine intelligence.

What the Hell is Concrete vs. Abstract Thinking? / Does anyone actually know?

This “post question” is vital to untangling much of what is said by “experts” about the ASD / Asperger “way of thinking”. One reads that we are “socially stupid” because we think concretely; receive language literally; fail to “comprehend” the gloriously sophisticated and complex use of “social language” (Have a nice day! Those jeans make you look skinny!) The assertion is that concrete thinking ranks as a “lower level” type of thinking on the grandiose pyramidal system of “human social development”, which has become the only development that “counts” toward being a “true” Homo sapiens. Other “experts” claim that we are “good at” abstract thinking; math and science and engineering, but this assumes that these activities are exclusively the product of abstract thinking! Far from it. 

We have to start somewhere! 

Abstract Objects

First published Thu Jul 19, 2001; substantive revision Mon Feb 13, 2017

It is widely supposed that every entity falls into one of two categories: Some are concrete; the rest abstract. The distinction is supposed to be of fundamental significance for metaphysics and epistemology. This article surveys a number of recent attempts to say how it should be drawn.

Here we are again: this “supposed distinction” is everywhere – but I find myself muttering, as I read various articles and papers, “What the Hell is this person talking about” when they refer to abstract thinking? I “get” formal thinking in math and other systems; the need to discover, set up, find an equation or formula that is “accurate” for all cases; a generalization that “matches” certain general conditions and provides for solutions and predictions. But the rest of “reality”?

What the Hell are people talking about? Human language itself seems to be a big part of the problem – this obsessional necessity to “chop up” a smooth experiential existence into a word salad. Yes, this is my Asperger confusion and frustration with “verbal language” – like using a chainsaw to carve butter. 


The Stanford Encyclopedia of Philosophy organizes scholars from around the world in philosophy and related disciplines to create and maintain an up-to-date reference work. Principal Editor: Edward N. Zalta

1. Introduction

The abstract/concrete distinction has a curious status in contemporary philosophy. It is widely agreed that the distinction is of fundamental importance. And yet there is no standard account of how it should be drawn. There is a great deal of agreement about how to classify certain paradigm cases. Thus it is universally acknowledged that numbers and the other objects of pure mathematics are abstract (if they exist), whereas rocks and trees and human beings are concrete. Some clear cases of abstracta are classes, propositions, concepts, the letter ‘A’, and Dante’s Inferno. Some clear cases of concreta are stars, protons, electromagnetic fields, the chalk tokens of the letter ‘A’ written on a certain blackboard, and James Joyce’s copy of Dante’s Inferno.

The challenge is to say what underlies this dichotomy, either by defining the terms explicitly, or by embedding them in a theory that makes their connections to other important categories more explicit. In the absence of such an account, the philosophical significance of the contrast remains uncertain. We may know how to classify things as abstract or concrete by appeal to intuition. But in the absence of theoretical articulation, it will be hard to know what (if anything) hangs on the classification.

Well, I’m not alone in my confusion!

It should be stressed that there need not be one single “correct” way of explaining the abstract/concrete distinction. Any plausible account will classify the paradigm cases in the standard way, and any interesting account will draw a clear and philosophically significant line in the domain of objects. Yet there may be many equally interesting ways of accomplishing these two goals, and if we find ourselves with two or more accounts that do the job rather well, there will be no point in asking which corresponds to the real abstract/concrete distinction. This illustrates a general point: when technical terminology is introduced in philosophy by means of examples but without explicit definition or theoretical elaboration, the resulting vocabulary is often vague or indeterminate in reference. In such cases, it is normally pointless to seek a single correct account. A philosopher may find himself asking questions like, ‘What isis idealism?’ or ‘What isis a substance?’ and treating these questions as difficult questions about the underlying nature of a certain determinate philosophical category. A better approach is to recognize that in many cases of this sort, we simply have not made up our minds about how the term is to be understood, and that what we seek is not a precise account of what this term already means, but rather a proposal for how it might fruitfully be used in the future. Anyone who believes that something in the vicinity of the abstract/concrete distinction matters for philosophy would be well advised to approach the project of explaining the distinction with this in mind.

2. Historical Remarks

The contemporary distinction between abstract and concrete is not an ancient one. Indeed, there is a strong case for the view that despite occasional anticipations, it played no significant role in philosophy before the 20th century. The modern distinction bears some resemblance to Plato’s distinction between Forms and Sensibles. But Plato’s Forms were supposed to be causes par excellence, whereas abstract objects are generally supposed to be causally inert in every sense. The original ‘abstract’/‘concrete’ distinction was a distinction among words or terms. Traditional grammar distinguishes the abstract noun ‘whiteness’ from the concrete noun ‘white’ without implying that this linguistic contrast corresponds to a metaphysical distinction in what these words stand for. In the 17th century this grammatical distinction was transposed to the domain of ideas. Locke speaks of the general idea of a triangle which is “neither Oblique nor Rectangle, neither Equilateral, Equicrural nor Scalenon [Scalene]; but all and none of these at once,” remarking that even this idea is not among the most “abstract, comprehensive and difficult” (Essay IV.vii.9). Locke’s conception of an abstract idea as one that is formed from concrete ideas by the omission of distinguishing detail was immediately rejected by Berkeley and then by Hume. But even for Locke there was no suggestion that the distinction between abstract ideas and concrete or particular ideas corresponds to a distinction among objects. “It is plain, …” Locke writes, “that General and Universal, belong not to the real existence of things; but are Inventions and Creatures of the Understanding, made by it for its own use, and concern only signs, whether Words or Ideas” (III.iii.11). (I agree)

The abstract/concrete distinction in its modern form is meant to mark a line in the domain of objects or entities. So conceived, the distinction becomes a central focus for philosophical discussion only in the 20th century. The origins of this development are obscure, but one crucial factor appears to have been the breakdown of the allegedly exhaustive distinction between the mental and the material that had formed the main division for ontologically minded philosophers since Descartes. One signal event in this development is Frege’s insistence that the objectivity and aprioricity of the truths of mathematics entail that numbers are neither material beings nor ideas in the mind. If numbers were material things (or properties of material things), the laws of arithmetic would have the status of empirical generalizations. If numbers were ideas in the mind, then the same difficulty would arise, as would countless others. (Whose mind contains the number 17? Is there one 17 in your mind and another in mine? In that case, the appearance of a common mathematical subject matter is an illusion.) In The Foundations of Arithmetic (1884), Frege concludes that numbers are neither external ‘concrete’ things nor mental entities of any sort. Later, in his essay “The Thought” (Frege 1918), he claims the same status for the items he calls thoughts—the senses of declarative sentences—and also, by implication, for their constituents, the senses of subsentential expressions. Frege does not say that senses are ‘abstract’. He says that they belong to a ‘third realm’ distinct both from the sensible external world and from the internal world of consciousness. Similar claims had been made by Bolzano (1837), and later by Brentano (1874) and his pupils, including Meinong and Husserl. The common theme in these developments is the felt need in semantics and psychology as well as in mathematics for a class of objective (i.e., non-mental) supersensible entities. As this new ‘realism’ was absorbed into English speaking philosophy, the traditional term ‘abstract’ was enlisted to apply to the denizens of this ‘third realm’.

Philosophers who affirm the existence of abstract objects are sometimes called platonists; those who deny their existence are sometimes called nominalists. This terminology is lamentable, since these words have established senses in the history of philosophy, where they denote positions that have little to do with the modern notion of an abstract object. However, the contemporary senses of these terms are now established, and so the reader should be aware of them. (In Anglophone philosophy, the most important source for this terminological innovation is Quine. See especially Goodman and Quine 1947.) In this connection, it is essential to bear in mind that modern platonists (with a small ‘p’) need not accept any of the distinctive metaphysical and epistemological doctrines of Plato, just as modern nominalists need not accept the distinctive doctrines of the medieval nominalists. Insofar as these terms are useful in a contemporary setting, they stand for thin doctrines: platonism is the thesis that there is at least one abstract object; nominalism is the thesis that the number of abstract objects is exactly zero (Field 1980). The details of this dispute are discussed in the article on nominalism in metaphysics. (See also the entry on platonism in metaphysics.) The aim of the present article is not to describe the case for or against the existence of abstract objects, but rather to say what an abstract object would be if such things existed.

3. The Way of Negation

Frege’s way of drawing the abstract/concrete distinction is an instance of what Lewis (1986a) calls the Way of Negation, according to which abstract objects are defined as those which lack certain features possessed by paradigmatic concrete objects. Nearly every explicit characterization in the literature follows this model. Let us review some of the options.

According to the account implicit in Frege’s writings, An object is abstract if and only if it is both non-mental and non-sensible.

Here the first challenge is to say what it means for a thing to be ‘non-mental’, or as we more commonly say, ‘mind-independent’. The simplest approach is to say that a thing depends on the mind when it would not (or could not) have existed if minds had not existed. But this entails that tables and chairs are mind-dependent, and that is not what philosophers who employ this notion have in mind. To call an object ‘mind-dependent’ in a metaphysical context is to suggest that it somehow owes its existence to mental activity, but not in the boring ‘causal’ sense in which ordinary artifacts owe their existence to the mind. What can this mean? One promising approach is to say that an object should be reckoned mind-dependent when, by its very nature, it exists at a time if and only if it is the object or content of some mental state or process at that time. This counts tables and chairs as mind-independent, since they might survive the annihilation of thinking things. But it counts paradigmatically mental items, like the purple afterimage of which I am now aware, as mind-dependent, since it presumably lies in the nature of such items to be objects of conscious awareness whenever they exist. However, it is not clear that this account captures the full force of the intended notion. Consider, for example, the mereological fusion of my afterimage and your headache. This is surely a mental entity if anything is. But it is not necessarily the object of a mental state. (The fusion can exist even if no one is thinking about itit .) A more generous conception would allow for mind-dependent objects that exist at a time in virtue of mental activity at that time, even if the object is not the object of any single mental state or act. The fusion of my afterimage plus your headache is mind-dependent in the second sense but not the first. That is a reason to prefer the second account of mind-dependence.

If we understand the notion of mind-dependence in this way, it is a mistake to insist that abstract objects be mind-independent. To strike a theme that will recur, it is widely supposed that sets and classes are abstract entities—even the impure sets whose urelements are concrete objects. Any account of the abstract/concrete distinction that places set-theoretic constructions like {{ Alfred, {{ Betty, {{ Charlie, Deborah}}}}}} on the concrete side of the line will be seriously at odds with standard usage. With this in mind, consider the set whose sole members are my afterimage and your headache, or some more complex set-theoretic object based on these items. If we suppose, as is plausible, that an impure set exists at a time only when its members exist at that time, this will be a mind-dependent entity in the generous sense. But it is also presumably an abstract entity. Gee whiz!

A similar problem arises for so-called abstract artifacts, like Jane Austen’s novels and the characters that inhabit them. Some philosophers regard such items as eternally existing abstract entities that worldly authors merely ‘describe’ or ‘encode’ but do not create. (Really?) But of course the commonsensical view is that Austen created Pride and Prejudice and Elizabeth Bennett, and there is no good reason to deny this (Thomasson 1999; cf. Sainsbury 2009; see also the entry on fiction). If we take this commonsensical approach, there will be a clear sense in which these items depend for their existence on Austen’s mental activity, and perhaps on the mental activity of subsequent readers. These items may not count as mind-dependent in either of the senses canvassed above, since Pride and Prejudice can presumably exist at a time even if no one happens to be thinking at that time. (If the world took a brief collective nap, Pride and Prejudice would not pop out of existence.) But they are obviously mind-dependent in some not-merely-causal sense. And yet they are still presumably abstract objects. For these reasons, it is probably a mistake to insist that abstract objects be mind-independent. (For more on mind-dependence, see Rosen 1994.)

Frege’s proposal in its original form also fails for other reasons. Quarks and electrons are neither sensible nor mind-dependent. And yet they are not abstract objects. A better version of Frege’s proposal would hold that:

An object is abstract if and only if it is both non-physical and non-mental.

This approach may well draw an important line; but it inherits the familiar problem of saying what it is for a thing to be a physical object (Crane and Mellor 1990). For discussion, see the entry on physicalism.

3.1 The Non-Spatiality Criterion

Contemporary purveyors of the Way of Negation typically amend Frege’s criterion by requiring that abstract objects be non-spatial, causally inefficacious, or both. Indeed, if any characterization of the abstract deserves to be regarded as the standard one, it is this:

An object is abstract if and only if it is non-spatial and causally inefficacious.

This standard account nonetheless presents a number of perplexities.

Consider first the requirement that abstract objects be non-spatial (or non-spatiotemporal). Some of the paradigms of abstractness are non-spatial in a straightforward sense. It makes no sense to ask where the cosine function was last Tuesday. Or if it makes sense to ask, the only sensible answer is that it was nowhere. Similarly, it makes no good sense to ask when the Pythagorean Theorem came to be. Or if it does make sense to ask, the only sensible answer is that it has always existed, or perhaps that it does not exist ‘in time’ at all. These paradigmatic ‘pure abstracta’ have no non-trivial spatial or temporal properties. They have no spatial location, and they exist nowhere in particular in time.

However, some abstract objects appear to stand in a more interesting relation to space. Consider the game of chess, for example. Some philosophers will say that chess is like a mathematical object, existing nowhere and ‘no when’—either eternally or outside of time altogether. But that is not the most natural view. The natural view is that chess was invented at a certain time and place (though it may be hard to say exactly where or when); that before it was invented it did not exist at all; that it was imported from India into Persia in the 7th century; that it has changed over the years, and so on. The only reason to resist this natural account is the thought that since chess is clearly an abstract object—it’s not a physical object, after all!—and since abstract objects do not exist in space and time—by definition!—chess must resemble the cosine function in its relation to space and time. And yet one might with equal justice regard the case of chess and other abstract artifacts as counterexamples to the hasty view that abstract objects possess only trivial spatial and temporal properties.

Should we then abandon the non-spatiotemporality criterion? Not necessarily. Even if there is a sense in which some abstract entities possess non-trivial spatiotemporal properties, it might still be said that concrete entities exist in spacetime in a distinctive way. If we had an account of this distinctive manner of spatiotemporal existence characteristic of concrete objects, we could say: An object is abstract (if and) only if it fails to exist in spacetime in that way.

One way to implement this approach is to note that paradigmatic concrete objects tend to occupy a relatively determinate spatial volume at each time at which they exist, or a determinate volume of spacetime over the course of their existence. It makes sense to ask of such an object, ‘Where is it now, and how much space does it occupy?’ even if the answer must sometimes be somewhat vague. By contrast, even if the game of chess is somehow ‘implicated’ in space and time, it makes no sense to ask how much space it now occupies. (To the extent that this does make sense, the only sensible answer is that it occupies no space at all, which is not to say that it occupies a spatial point.) And so it might be said:

An object is abstract (if and) only if it fails to occupy anything like a determinate region of space (or spacetime).

This promising idea raises several questions. First, it is conceivable that certain items that are standardly regarded as abstract might nonetheless occupy determinate volumes of space and time. Consider, for example, the various sets composed from Peter and Paul: {{ Peter, Paul},{},{ Peter, {{ Peter, {{{{ Paul}}}}}}}} , etc. We don’t normally ask where such things are, or how much space they occupy. And indeed many philosophers will say that the question makes no sense, or that the answer is a dismissive ‘nowhere, none’. But this answer is not forced upon us by anything in set theory or metaphysics. Even if we grant that pure sets stand in only the most trivial relations to space, it is open to us to hold, as some philosophers have done, that impure sets exist where and when their members do (Lewis 1986a). It is not unnatural to say that a set of books is located on a certain shelf in the library, and indeed, there are some theoretical reasons for wanting to say this (Maddy 1990). On a view of this sort, we face a choice: we can say that since impure sets exist in space, they are not abstract objects after all; or we can say that since impure sets are abstract, it was a mistake to suppose that abstract objects cannot occupy space.

One way to finesse this difficulty would be to note that even if impure sets occupy space, they do so in a derivative manner. The set {{ Peter, Paul}} occupies a location in virtue of the fact that its concrete elements, Peter and Paul, together occupy that location. The set does not occupy the location in its own right. With that in mind, it might be said that:

An object is abstract (if and) only if it either fails to occupy space at all, or does so only in virtue of the fact some other items—in this case, its urelements—occupy that region.

But of course Peter himself occupies a region in virtue of the fact that his parts—his head, hands, etc.—together occupy that region. So a better version of the proposal would say:

An object is abstract (if and) only if it either fails to occupy space at all, or does so of the fact that some other items that are not among its parts occupy that region.

This approach appears to classify the cases fairly well, but it is somewhat artificial. Moreover it raises a number of questions. What are we to say about the statue that occupies a region of space, not because its parts are arrayed in space, but rather because its constituting matter occupies that region? And what about the unobserved electron, which according to some interpretations of quantum mechanics does not really occupy a region of space at all, but rather stands in some more exotic relation to the spacetime it inhabits? Suffice it to say that a philosopher who regards ‘non-spatiality’ as a mark of the abstract, but who allows that some abstract objects may have non-trivial spatial properties, owes us an account of the distinctive relation to space and spacetime that sets paradigmatic concreta apart.

Perhaps the most important question about the ‘non-spatiality’ criterion concerns the classification of the parts of space itself. Let us suppose that space or spacetime exists, not just as an object of pure mathematics, but as the arena in which physical objects and events are somehow arrayed. Physical objects are located ‘in’ or ‘at’ regions of space, and so count as concrete according to the non-spatiality criterion. But what about the points and regions of space itself? There has been some debate about whether a commitment to spacetime substantivalism is consistent with the nominalist’s rejection of abstract entities (Field 1980, 1989; Malament 1982). If we define the abstract as the ‘non-spatial’, this debate reduces to the question whether space itself is to be reckoned ‘spatial’. But surely that is a verbal question. We can extend existing usage so as to allow that points and regions of space are located ‘at’ themselves—or not, according to taste. The philosopher who thinks that there is a serious question about whether the parts of space count as concrete would thus do well to characterize the abstract/concrete distinction in other terms.

3.2 The Causal Inefficacy Criterion

According to the most widely accepted versions of the Way of Negation:

An object is abstract (if and) only if it is causally inefficacious.

Concrete objects, whether mental or physical, have causal powers; numbers and functions and the rest make nothing happen. There is no such thing as causal commerce with the game of chess itself (as distinct from its concrete instances). And even if impure sets do in some sense exist in space, it is easy enough to believe that they make no distinctive causal contribution to what transpires. Peter and Paul may have effects individually. They may even have effects together that neither has on his own. But these joint effects are naturally construed as effects of two concrete objects acting jointly, or perhaps as effects of their mereological aggregate (itself a paradigm concretum), rather than as effects of some set-theoretic construction. Suppose Peter and Paul together tip a balance. If we entertain the possibility that this event is caused by a set, we shall have to ask which set caused it: the set containing just Peter and Paul? Some more elaborate construction based on them? Or is it perhaps the set containing the molecules that compose Peter and Paul? This proliferation of possible answers suggests that it was a mistake to credit sets with causal powers in the first place. This is good news for those who wish to say that all sets are abstract.

(Note, however, that some writers identify ordinary physical events—causally efficacious items par excellence—with sets. For David Lewis, for example, an event like the fall of Rome is an ordered pair whose first member is a region of spacetime, and whose second member is a set of such regions (Lewis 1986b). On this account, it would be disastrous to say both that impure sets are abstract objects, and that abstract objects are non-causal.)

The idea that causal inefficacy constitutes a sufficient condition for abstractness is somewhat at odds with standard usage. Some philosophers believe in ‘epiphenomenal qualia’: objects of conscious awareness (sense data), or qualitative conscious states that may be caused by physical processes in the brain, but which have no downstream causal consequences of their own (Jackson 1982; Chalmers 1996). These items are causally inefficacious if they exist, but they are not normally regarded as abstract. The proponent of the causal inefficacy criterion might respond by insisting that abstract objects are distinctively neither causes nor effects. But this is perilous. Abstract artifacts like Jane Austen’s novels (as we normally conceive them) come into being as a result of human activity. The same goes for impure sets, which come into being when their concrete urelements are created. These items are clearly effects in some good sense; yet they remain abstract if they exist at all. It is unclear how the proponent of the strong version of the causal inefficacy criterion (which views causal inefficacy as both necessary and sufficient for abstractness) might best respond to this problem.

Apart from this worry, there are no decisive intuitive counterexamples to this account of the abstract/concrete distinction. The chief difficulty—and it is hardly decisive—is rather conceptual. It is widely maintained that causation, strictly speaking, is a relation among events or states of affairs. If we say that the rock—an object—caused the window to break, what we mean is that some event or state (or fact or condition) involving the rock caused the break. If the rock itself is a cause, it is a cause in some derivative sense. But this derivative sense has proved elusive. The rock’s hitting the window is an event in which the rock ‘participates’ in a certain way, and it is because the rock participates in events in this way that we credit the rock itself with causal efficacy. But what is it for an object to participate in an event? Suppose John is thinking about the Pythagorean Theorem and you ask him to say what’s on his mind. His response is an event—the utterance of a sentence; and one of its causes is the event of John’s thinking about the theorem. Does the Pythagorean Theorem ‘participate’ in this event? There is surely some sense in which it does. The event consists in John’s coming to stand in a certain relation to the theorem, just as the rock’s hitting the window consists in the rock’s coming to stand in a certain relation to the glass. But we do not credit the Pythagorean Theorem with causal efficacy simply because it participates in this sense in an event which is a cause. The challenge is therefore to characterize the distinctive manner of ‘participation in the causal order’ that distinguishes the concrete entities. This problem has received relatively little attention. There is no reason to believe that it cannot be solved. But in the absence of a solution, this standard version of the Way of Negation must be reckoned a work in progress.

4. The Way of Example

In addition to the Way of Negation, Lewis identifies three main strategies for explaining the abstract/concrete distinction. According to the Way of Example, it suffices to list paradigm cases of abstract and concrete entities in the hope that the sense of the distinction will somehow emerge. If the distinction were primitive and unanalyzable, this might be the only way to explain it. But as we have remarked, this approach is bound to call the interest of the distinction into question. The abstract/concrete distinction matters because abstract objects as a class appear to present certain general problems in epistemology and the philosophy of language. It is supposed to be unclear how we come by our knowledge of abstract objects in a sense in which it is not unclear how we come by our knowledge of concrete objects (Benacerraf 1973). It is supposed to be unclear how we manage to refer determinately to abstract entities in a sense in which it is not unclear how we manage to refer determinately to other things (Benacerraf 1973, Hodes 1984). But if these are genuine problems, there must be some account of why abstract objects as such should be especially problematic in these ways. It is hard to believe that it is simply their primitive abstractness that makes the difference. It is much easier to believe that it is their non-spatiality or their causal inefficacy or something of the sort. It is not out of the question that the abstract/concrete distinction is fundamental, and that the Way of Example is the best we can do by way of elucidation. But if so, it is quite unclear why the distinction should make a difference.

5. The Way of Conflation

According to the Way of Conflation, the abstract/concrete distinction is to be identified with one or another metaphysical distinction already familiar under another name: as it might be, the distinction between sets and individuals, or the distinction between universals and particulars. There is no doubt that some authors have used the terms in this way. (Thus Quine 1953 uses ‘abstract entity’ and ‘universal’ interchangeably.) This sort of conflation is however rare in recent philosophy.

6. The Way of Abstraction

The most important alternative to the Way of Negation is what Lewis calls the Way of Abstraction. According to a longstanding tradition in philosophical psychology, abstraction is a distinctive mental process in which new ideas or conceptions are formed by considering several objects or ideas and omitting the features that distinguish them. For example, if one is given a range of white things of varying shapes and sizes; one ignores or ‘abstracts from’ the respects in which they differ, and thereby attains the abstract idea of whiteness. Nothing in this tradition requires that ideas formed in this way represent or correspond to a distinctive kind of object. But it might be maintained that the distinction between abstract and concrete objects should be explained by reference to the psychological process of abstraction or something like it. The simplest version of this strategy would be to say that an object is abstract if it is (or might be) the referent of an abstract idea, i.e., an idea formed by abstraction.

So conceived, the Way of Abstraction is wedded to an outmoded philosophy of mind. But a related approach has gained considerable currency in recent years. Crispin Wright (1983) and Bob Hale (1987) have developed an account of abstract objects that takes leave from certain suggestive remarks in Frege (1884). Frege notes (in effect) that many of the singular terms that appear to refer to abstract entities are formed by means of functional expressions. We speak of the shape of a building, the direction of a line, the number of books on the shelf. Of course many singular terms formed by means of functional expressions denote ordinary concrete objects: ‘the father of Plato’, ‘the capital of France’. But the functional terms that pick out abstract entities are distinctive in the following respect: Where ‘f(a)f(a) ’ is such an expression, there is typically an equation of the form

f(a)=f(b) if and only if Rab,f(a)=f(b) if and only if Rab,where RR is an equivalence relation. (An equivalence relation is a relation that is reflexive, symmetric and transitive.)
For example:

The direction of a=a= the direction of bb if and only if aa is parallel to bb .

The number of FF s = the number of GG s if and only if there are just as many FF s as GG s.

Moreover, these equations (or abstraction principles) appear to have a special semantic status. While they are not strictly speaking definitions of the functional expression that occurs on the left hand side, they would appear to hold in virtue of the meaning of that expression. To understand the term ‘direction’ is (in part) to know that ‘the direction of aa ’ and ‘the direction of bb ’ refer to the same entity if and only if the lines aa and bb are parallel. Moreover, the equivalence relation that appears on the right hand side of the equation would appear to be semantically and perhaps epistemologically prior to the functional expression on the left (Noonan 1978). Mastery of the concept of a direction presupposes mastery of the concept of parallelism, but not vice versa.

The availability of abstraction principles meeting these conditions may be exploited to yield an account of the distinction between abstract and concrete objects. When ‘ff ’ is a functional expression governed by an abstraction principle, there will be a corresponding kind KfKf such that:

x is a Kf if and only if, for some y,x=f(y).x is a Kf if and only if, for some y,x=f(y).

For example, xx is a cardinal number if and only if for some concept F,x=F,x= the number of FsFs . The simplest version of this approach to the Way of Abstraction is then to say that

xx is an abstract object if (and only if) xx is an instance of some kind KfKf whose associated functional expression ‘ff ’ is governed by a suitable abstraction principle.

The strong version of this account—which purports to identify a necessary condition for abstractness—is seriously at odds with standard usage. As we have noted, pure sets are paradigmatic abstract objects. But it is not clear that they satisfy the proposed criterion. According to naïve set theory, the functional expression ‘set of’ is indeed characterized by a putative abstraction principle.

The set of FF s = the set of GG s if and only if, for all x,(xx,(x is FF if and only if xx is G)G) .

But this principle is inconsistent, and so fails to characterize an interesting concept. In contemporary mathematics, the concept of a set is not introduced by abstraction. It remains an open question whether something like the mathematical concept of a set can be characterized by a suitably restricted abstraction principle. (See Burgess 2005 for a survey of recent efforts in this direction.) Even if such a principle is available, however, it is unlikely that the epistemological priority condition will be satisfied. (That is, it is unlikely that mastery of the concept of set will presuppose mastery of the equivalence relation that figures on the right hand side.) It is therefore uncertain whether the Way of Abstraction so understood will classify the objects of pure set theory as abstract entities (as it presumably must).

Similarly, as Dummett (1973) has noted, in many cases the standard names for paradigmatically abstract objects do not assume the functional form to which the definition adverts. Chess is an abstract entity. But we do not understand the word ‘chess’ as synonymous with an expression of the form ‘f(x)f(x) ’ where ‘ff ’ is governed by an abstraction principle. Similar remarks would seem to apply to such things as the English language, social justice, architecture, and Charlie Parker’s style. If so, the abstractionist approach does not provide a necessary condition for abstractness as that notion is standardly understood.

More importantly, there is some reason to believe that it fails to supply a sufficient condition. A mereological fusion of concrete objects is itself a concrete object. But the concept of a mereological fusion is governed by what appears to be an abstraction principle:

The fusion of the FF s = the fusion of the GG s if and only if the FF s and GG s cover one another,

where the FF s cover the GG s if and only if every part of every GG has a part in common with an FF . Similarly, suppose a train is a maximal string of railroad carriages, all of which are connected to one another. We may define a functional expression, ‘the train of xx ’, by means of an ‘abstraction’ principle: The train of x=x= the train of yy iff (if and only if) xx and yy are connected carriages. We may then say that xx is a train iff for some carriage yy , xx is the train of yy . The simple account thus yields the consequence that trains are to be reckoned abstract entities.

It is unclear whether these objections apply to the more sophisticated abstractionist proposals of Wright and Hale, but one feature of the simple account sketched above clearly does apply to these proposals and may serve as the basis for an objection to this version of the Way of Abstraction. The neo-Fregean approach seeks to explain the abstract/concrete distinction in semantic terms: We said that an abstract object is an object that falls in the range of a functional expression governed by an abstraction principle, where ‘ff ’ is governed by an abstraction principle when that principle holds in virtue of the meaning of ‘ff ’. This notion of a statement’s holding in virtue of the meaning of a word is notoriously problematic (see the entry on the analytic-synthetic distinction). But even if this notion makes sense, one may still complain: The abstract/concrete distinction is supposed to be a metaphysical distinction; abstract objects are supposed to differ from other objects in some important ontological respect. It should be possible, then, to draw the distinction directly in metaphysical terms: to say what it is in the objects themselves that makes some things abstract and others concrete. As Lewis writes, in response to a related proposal by Dummett:

Even if this … way succeeds in drawing a border, as for all I know it may, it tells us nothing about how the entities on opposite sides of that border differ in their nature. It is like saying that snakes are the animals that we instinctively most fear—maybe so, but it tells us nothing about the nature of snakes. (Lewis 1986a: 82)

The challenge is to produce a non-semantic version of the abstractionist criterion that specifies directly, in metaphysical terms, what the objects whose canonical names are governed by abstraction principles all have in common.

One response to this difficulty is to transpose the abstractionist proposal into more metaphysical key. We begin with the idea that each Fregean number is, by its very nature, the number of some Fregean concept, just as each Fregean direction is, by its very nature, at least potentially the direction of some concrete line. In each case, the abstract object is essentially the value of an abstraction function for a certain class of arguments. This is not a claim about the meanings of linguistic expressions. It is a claim about the essences or natures of the objects themselves. (For the relevant notion of essence, see Fine 1994). So for example, the Fregean number two (if there is such a thing) is, essentially, by its very nature, the number that belongs to a concept FF if and only if there are exactly two FF s. More generally, for each Fregean abstract object xx , there is an abstraction function ff , such that xx is essentially the value of ff for every argument of a certain kind.

Abstraction functions have two key features. First, for each abstraction function ff there is an equivalence relation RR such that it lies in the nature of ff that f(x)=f(y)f(x)=f(y) iff Rxy. Intuitively, we are to think that RR is metaphysically prior to ff , and that the abstraction function ff is defined (in whole or in part) by this biconditional. Second, each abstraction function is a generating function: its values are essentially values of that function. Many functions are not generating functions. Paris is the capital of France, but it is not essentially a capital. The number of solar planets, by contrast, is essentially a number. The notion of an abstraction function may be defined in terms of these two features:

  • ff is an abstraction function iff
  • a.for some equivalence relation RR , it lies in the nature of ff that f(x)=f(y)f(x)=f(y) iff RxyRxy ; and
  • b.for all xx , if xx is a value of ff , then it lies in the nature of xx that there is (or could be) some object yy such that x=f(y)x=f(y) .

We may then say that

xx is an abstraction if and only if, for some abstraction function ff , there is or could be an object yy such that x=f(y)x=f(y)


xx is an abstract object if (and only if) xx is an abstraction.

This account tells us a great deal about the distinctive natures of these broadly Fregean abstract objects. It tells us that each is, by its very nature, the value of a special sort of function, one whose nature is specified in a simple way in terms of an associated equivalence relation. It is worth stressing, however, that it does not supply much metaphysical information about these items. It does tell us whether they are located in space, whether they can stand in causal relations, and so on. It is an open question whether this somewhat unfamiliar version of the abstract/concrete distinction lines up with any of the more conventional ways of drawing the distinction outlined above.

7. Further Reading

Putnam (1975) makes the case for abstract objects on scientific grounds. Field (1980, 1989) makes the case against abstract objects. Bealer (1993) and Tennant (1997) present a priori arguments for the necessary existence of abstract entities. Balaguer (1998) argues that none of the arguments for or against the existence of abstract objects is compelling, and that there is no fact of the matter as to whether abstract things exist. The dispute over the existence of abstracta is reviewed in Burgess and Rosen (1997). Fine (2002) is a systematic study of abstraction principles in the foundations of mathematics. A general theory of abstract objects is developed axiomatically in Zalta (1983; 2016 in Other Internet Resources). Wetzel (2009) examines the type-token distinction, argues that types are abstract objects while the tokens of those types are their concrete instances, and shows how difficult it is to paraphrase away the many references to types that occur in the sciences and natural language. (See the entry on types and tokens.) Moltmann (2013) investigates the extent to which abstract objects are needed when developing a semantics of natural language.

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