I suspect both philosophers and mathematicians succeed and fail on this issue in a wide range of degrees.
I'm not sure what you mean by "fail on this issue". Are you saying that mathematicians who wonder about the physical realizability of a Hilbert-Hotel type scenario using the NASC definition of infinity are committing the error of which you, Lakoff, and Johnson accuse conceptual-analysis style philosophy? (You probably don't mean that, but I'm giving you my best guess so that you can bounce off of it while clarifying your intended meaning.)
Sorry, I'm not familiar enough with the studies of infinity to say.
In my last post, I showed that the brain does not encode concepts in terms of necessary and sufficient conditions. So, any philosophical practice which assumes this — as much of 20th century conceptual analysis seems to do — is misguided.
Next, I want to show that human abstract thought is pervaded by metaphor, and that this has implications for how we think about the nature of philosophical questions and philosophical answers. As Lakoff & Johnson (1999) write:
To understand how fundamental metaphor is to our thinking, we must remember that human cognition is embodied:
Consider, for example, the fact that as neural beings we must categorize things:
Moreover, almost all our categorizations are determined by the unconscious associative mind — outside our control and even our awareness — as we interact with the world. As Lakoff & Johnson note, "Even when we think we are deliberately forming new categories, our unconscious categories enter into our choice of possible conscious categories."
And because our categories are shaped not by a transcendent, universal faculty of reason but by the components of our sensorimotor system that process our interaction with the world, our concepts and categories end up being largely sensorimotor concepts and categories.
Here are some examples of metaphorical thought shaped by the sensorimotor system:
Important Is Big
Example: "Tomorrow is a big day."
Mapping: From importance to size.
Experience: As a child, finding that big things (e.g. parents) are important and can exert major forces on you and dominate your visual experience.
Intimacy Is Closeness
Example: "We've been close for years, but we're beginning to drift apart."
Mapping: From intimacy to physical proximity.
Experience: Being physically close to people you are intimate with.
Difficulties Are Burdens
Example: "She's weighed down by her responsibilities."
Mapping: From difficulty to muscular exertion.
Experience: The discomfort or disabling effect of lifting or carrying heavy objects.
More Is Up
Example: "Prices are high."
Mapping: From quantity to vertical orientation.
Experience: Observing the rise and fall of levels of piles and fluids as more is added or subtracted.
Categories Are Containers
Example: "Are tomatoes in the fruit or vegetable category?"
Mapping: From kinds to spatial location.
Experience: Observing that things that go together tend to be in the same bounded region.
Linear Scales Are Paths
Example: "John's intelligence goes way beyond Bill's."
Mapping: From degree to motion in space.
Experience: Observing the amount of progress made by an object.
Organization Is Physical Structure
Example: "How do the pieces of this theory fit together?"
Mapping: From abstract relationships to experience with physical objects.
Experience: Interacting with complex objects and attending to their structure.
States Are Locations
Example: "I'm close to being in a depression and the next thing that goes wrong will send me over the edge.
Mapping: From a subjective state to being in a bounded region of space.
Experience: Experiencing a certain state as correlated with a certain location (e.g. being cool under a tree, feeling secure in a bed).
Purposes Are Destinations
Example: "He'll ultimately be successful, but he isn't there yet."
Mapping: From achieving a purpose to reaching a destination in space.
Experience: Reaching destinations throughout everyday life and thereby achieving purposes (e.g. if you want food, you have to go to the fridge).
Actions Are Motions
Example: "I'm moving right along on the project."
Mapping: From action to moving your body through space.
Experience: The common action of moving yourself through space, especially in the early years of life when that is to some degree the only kind of action you can take.
Understanding Is Grasping
Example: "I've never been able to grasp transfinite numbers."
Mapping: From comprehension to object manipulation.
Experience: Getting information about an object by grasping and manipulating it.
As a neural being interacting with the world, you can't help but build up such "primary" metaphors:
Primary metaphors are combined to build complex metaphors. For example, Actions Are Motions and Purposes Are Destinations are often combined to form a new metaphor:
A Purposeful Life is a Journey
Example: "She seems lost, without direction. She's fallen off track. She needs to find her purpose and get moving again."
Can we think without metaphor, then? Yes. Our concepts of so-called "basic level" objects (that we interact with in everyday experience) are often literal, as are sensorimotor concepts. Our concepts of "tree" (the thing that grows in dirt), "grasp" (holding an object), and "in" (in the spatial sense) are all literal. But when it comes to abstract reasoning or subjective judgment, we tend to think in metaphor. We can't help it.
Implications for philosophical method
What happens when we fail to realize that our thinking is metaphorical? Let's consider a famous example: Zeno's paradox of the arrow.
Zeno described time as a sequence of points along a timeline. Now, consider an arrow in flight. At any point on the timeline, the arrow is at some particular fixed location. At a later point on the timeline, the arrow is at a different location. But since the arrow is located at a single fixed place at every point in time, then where is the motion?
The puzzle arises when you take the metaphor of time as discrete points along the space of a timeline as being literal:
Moral concepts as metaphors
For a more detailed illustration of the philosophical implications of metaphorical thought, let's examine the metaphors that ground our moral concepts:
Well-Being Is Wealth is not the only metaphor behind our moral thinking. Here are a few others:
Being Moral Is Being Upright; Being Immoral Is Being Low; Evil Is a Force
Example: "He's an upstanding citizen. She's on the up and up. She's as upright as they come. That was a low thing to do. He's underhanded. I would never stoop to such a thing. She fell from grace. She succumbed to the floods of emotion and the fires of passion. She didn't have enough moral backbone to stand up to evil."
How does the metaphorical nature of our moral concepts constrain moral philosophy? Let us contrast a traditional view of moral concepts with the view of moral concepts emerging from cognitive science:
An explosion of productivity in moral psychology since Lakoff & Johnson's book was published has confirmed these claims. The convergence of evidence suggests that multiple competing systems contribute to our moral reasoning, and they engage many processes not unique to moral reasoning.
Once again, knowledge of cognitive science constrains philosophy:
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