The short answer is -- "No!"
(A note of clarification: I will assume here some form of metaphysical realism. If the metaphysical premise were some form of idealism or phenominalism, then either it is highly questionable whether there could be something that both of us see and therefore could conceivably see differently, or it is highly questionable whether the fact that we do see it differently would be as problematic as the title question suggests.)
What would it mean for you to see the tomato as green and the green pepper as red while at the same time calling the tomato "red" and the green pepper "green"? For that to be the case, there would have to be two conditions met. One is physical (what you see), and one is epistemological (what you say or think or believe about what you see).
The physical condition is the easier to understand, more or less. At some point in the neural network connecting the retina to the consciousness, at the level of inter-nerve electro-chemical impulses, you would have to respond to a ripe tomato the way that I respond to a green pepper. (And vice versa of course, as stipulated.) It does not matter where in the neural net you propose to locate the difference between us, so let's suppose, for convenience, your difference is located in your retina. The rods and cone cells of your retina send nerve impulses up the optic nerve reporting on (among other things) the energy level (frequency) of the photons that strike them. Suppose that scientists have tapped into my optic nerve and recorded the nerve impulses that my retina sends to my brain when presented with either a red or a green visual field. Lets call the nerve pulse string that my retina sends to my brain for a red visual field "string(r)", and for a green visual field "string(g)". The proposal then is that when the scientists measure the nerve pulse strings on your optic nerve, they will detect string(r) when you are presented with a green visual field, and string(g) when you are presented by a red visual field. Physically, I think that this would be a reasonable sounding interpretation of "the things you see as red I see as green and vice versa."
It seems physically a rather simple proposal. But it does involve a couple of problematic presumptions. First of all, it means that we are assuming that any given nerve pulse string transmitted by my retina means the same to your brain as it does to mine. And since we picked the retina purely for intuitive convenience, it means that we are assuming that any given nerve pulse string measured anywhere in my process of seeing means the same to your brain as it does to mine -- regardless of where in the "seeing" process we choose to measure nerve-pulse strings. And second we are assuming that the process of seeing things can in fact be discerned at meaningful points as (reduced to) nerve-pulse strings. Along the optic nerve, maybe. But in other places? Perhaps any given brain's nerve-pulse strings are idiosyncratic consequences of the developmental and experiential histories of the brain involved. And even if it should turn out that your retina does transmit string(g) when shown a red visual field, and string(r) when shown a green visual field, I do not think it is a foregone conclusion that those messages mean "green" and "red" to your brain. I don't think either of these assumptions is reasonable at this point in the development of brain science. They should certainly not be taken for granted.
The epistemological condition is even more problematic. What would the words "red" and "green" actually mean to you if your electro-chemical nerve responses were actually as described above? "Red" and "Green" are the labels we each attach to two distinct collection concepts. Each of us forms our own concepts, and attaches to them suitable word tags as we learn to use a language. And the title question stipulates that we speak the same language.
For me, the concept "red" subsumes all of those existents in the world that I would classify as similar in that they appear to share a particular narrow range of colours. For me, the concept "green" subsumes all of those existents in the world that I would classify as similar in that they also appear to share a particular narrow range of colours, although a different range from that for my concept of "red". I learned these concepts ostensibly, by being shown examples of things that other people call red and green. To me "red" means all those things (or parts of things) that share this particular narrow range of colours.
By saying that we use the same language to describe things, the title question stipulates the constraint that we subsume the same set of existents within closely similar conceptual classifications, and have tagged those two classifications with the same words. In other words, by stipulation, everything that I call red you call red, and everything that I call green you call green. (We'll ignore the instances of boundary examples -- instances of "almost-red" or "almost green".) If that is the case, then "red" means all those things that we respectively call "red". And similarly for "green". What then would it mean to suggest that you see the tomato as green, when you subsume the tomato under your concept of "red" and speak of it as "red"? Even if your neural network responded with string(g) to the tomato and string(r) to the green pepper, this would not demonstrate that you see the tomato as green (and vice versa). The only possible interpretation of the nerve impulse strings is that, to your brain, string(g) means "red", and string(r) means "green".
Clearly, by hypothesis, you do not subsume the tomato under the same concept as you do the green pepper. And clearly, also by hypothesis, you subsume the tomato under a concept that you have tagged with the word "red", and not under the concept that you have tagged with the word "green". So how could you possibly understand the notion of seeing it as "green"? The hypothesized scenario is simply not meaningful.
The title question only appears to be intelligible because it assumes an indirect theory of perception, appeals to a hidden assumption of mind-body dualism, and commits two fallacies of equivocation on the meaning of perception.
Drawing upon the many examples where appearances are clearly at odds with the way reality really is, theories of indirect perception posit an intermediate between reality and appearance. The most popular of the indirect theories of perception is the "sense-data" theory. The notion is that our senses extract "sense-data" from reality which are then presented to the conscious awareness for appreciation and interpretation. The sense-data show us the appearance of reality. This model of perception allows that reality can be different from appearance. The three primary arguments presented in support of the premise that perception is in-direct are:
If indirect theories of perception are right, then there is something between the tomato and the conscious mind that sees the tomato. That thing is the "appearance" or "sense-data" of the tomato. The sense-data is invested with all of the properties possessed by the appearance of the tomato, and it is those properties that we actually perceive. Somehow, somewhere between the real object and our seeing of the object some intermediate medium or our sensory input processors "translate" the actual straightness of the stick in the water glass into a bent appearance, and "translate" the actual round shape of the table into an oval appearance, or generate a pink elephant appearance out of thin air. Because it is assumed that either some intervening medium or the sensory pathways of our nervous system do some processing of the data received directly from the real object before the "mind" gets to "see" it, it is therefore readily conceivable that the appearance of a stick (or a table, or thin air, or a tomato, or a green pepper) that is presented to the mind can be quite different from its actual "real" properties. With this image in mind, it now seems quite intelligible that the appearance of a red tomato might actually be "translated" into green, and vice versa for the green pepper. After all, if so much can be changed twixt the reality and the appearance, and if at least some of those changes are internal to our brains (somewhere between the retina and the mind), why cannot your internal sensory processes change things in a way that is different from the way my internal sensory processes change things? Hence the title question.
However, indirect theories of perception (at least those as "naive" as is required to permit the hypothesized spectral inversion differences between us) simply do not work. First of all, the sense-data theories assume that if something appears to have some property, there must exist something that has that property. But as Austin famously pointed out, this is not supportable. There is no need to reify appearances. If a church looks like a barn, we do not perceive a barn appearance. We perceive a church that looks like a barn. The appearance of a thing is just how we perceive it from here, under these conditions. The arguments from variation and illusion commit the fallacy of equivocation between the "dispositional" use versus the "sensible" use of property words. The dispositional use of properties describes the way that things are in fact. The sensible use describes the way that things appear. From the fact that some thing's appearance is a relational consequence of the conjunction of subject and object (and perhaps some intermediate medium), it is a fallacious move to conclude that the way the things are in fact is also a relational property. They also commit the fallacy of equivocation between the "first-person" versus "third-person" sense of the meaning of perceive (or sense, observe, see, smell, hear, touch, etc.) when describing what is going on. The third-person sense is us describing S's perception based on what we know here and now. The first-person sense is S describing S's perception based on what S knows there and then. Consider how we think about John and Jane seeing a "cat on the roof". Suppose John recognizes the black blob on the roof as his pet cat, while Jane only sees a black blob on the roof and does not recognize it as a cat. In both cases we say that John or Jane "sees a cat on the roof". But in John's case it is first-person, while in Jane's case it is third-person.
Secondly, if sense-data do have the properties they appear to have (which, according to the theory they must), where are these sense-data? And what are they? They are certainly not physical objects. And they certainly are not located in physical space in the place that they appear to be located. There are not two tables located over there, only one. Wherever sense data are located, if they are not physical objects, how do they come to have the properties they appear to have? If they are located in the head/brain/mind, again where are they? Of course, they may be simply encoded representations of what they appear to be -- brain-states of encoded neural network impulse strings. But if that is the case, in what sense can the sense-data be said to "have" the properties they are supposed to have? Moreover, sometimes it happens that our perceptions are somewhat indeterminate. There is an advertisement on the side of the building opposite. Yet if I take off my glasses, the writing is blurred and indeterminate. Is the sense-data of that advertisement determinate or indeterminate? And what would it mean for the sense-data to have the properties I am perceiving, if I can't read that blurry writing.
Which brings us to the third problem for indirect theories of perception. Such theories lean very heavily on an intuitive acceptance of Cartesian Dualism. Such theories demand somewhere in the mind/brain a place where sense-data ("appearances") can be presented for our conscious mind to appreciate and evaluate. They assume some place where the sense-data of a bent stick is to be interpreted and conceptualized. The brain processes and translates the sensory inputs into representations of the appearance of things. And the mind sees the appearance. Yet there is no indication from either brain science or cognitive science that there is such a place. At best, if the sense-data are in the head, as brain-states of encoded neural network impulse strings, then they are distributed throughout the brain. There is no place in the brain for a "higher" level of awareness that can appreciate this "appearance". No homunculus driving the truck. Unless you accept that the conscious awareness is not in the brain. And thus comes the title question. If the mind is separate from the brain, then perhaps your brain presents the tomato to your mind with a different appearance than my brain presents the tomato to my mind.
However, if you reject mind-body dualism, the only viable alternative (other than falling back to idealist or phenominalist theories, within which the title question would not make any sense in a more metaphysical way), is some form of direct realism. Given a direct realist theory of perception, the way that you see the tomato is the way that the tomato looks to you from this perspective, under these conditions. If the colour that the tomato appears to you as is "red", and you call it "red" when speaking about its colour, then there is no possibility that you could be seeing it as "green". (And vice versa for the green pepper, of course.) There is just no conceptual place for the "spectrum inversion" to take place.
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